We consider the problem of sampling from the ferromagnetic $q$-state Potts model on the random $d$-regular graph with parameter $β>0$. A key difficulty that arises in sampling from the model is the existence of a metastability window $(β_u,β_u')$ where the distribution has two competing modes, the so-called disordered and ordered phases, causing MCMC-based algorithms to be slow mixing from worst-case initialisations. To this end, Helmuth, Jenssen and Perkins designed a sampling algorithm that works for all $β$ when $q$ is large, using cluster expansion methods; more recently, their analysis technique has been adapted to show that random-cluster dynamics mixes fast when initialised more judiciously. However, a bottleneck behind cluster-expansion arguments is that they inherently only work for large $q$, whereas it is widely conjectured that sampling is possible for all $q,d\geq 3$. The only result so far that applies to general $q,d\geq 3$ is by Blanca and Gheissari who showed that the random-cluster dynamics mixes fast for $β<β_u$. For $β>β_u$, certain correlation phenomena emerge because of the metastability which have been hard to handle, especially for small $q$ and $d$. Our main contribution is to perform a delicate analysis of the Potts distribution and the random-cluster dynamics that goes beyond the threshold $β_u$. We use planting as the main tool in our proofs, and combine it with the analysis of random-cluster dynamics. We are thus able to show that the random-cluster dynamics initialised from all-out mixes fast for all integers $q,d\geq 3$ beyond the uniqueness threshold $β_u$; our analysis works all the way up to the threshold $β_c\in (β_u,β_u')$ where the dominant mode switches from disordered to ordered. We also obtain an algorithm in the ordered regime $β>β_c$ that refines significantly the range of $q,d$.

Consider a $k$-SAT formula $Φ$ where every variable appears at most $d$ times. Let $σ$ be a satisfying assignment, sampled proportionally to $e^{βm(σ)}$ where $m(σ)$ is the number of true variables and $β$ is a real parameter. Given $Φ$ and $σ$, can we efficiently learn $β$? This problem falls into a recent line of work about single-sample (``one-shot'') learning of Markov random fields. Our $k$-SAT setting was recently studied by Galanis, Kalavasis, Kandiros (SODA24). They showed that single-sample learning is possible when roughly $d\leq 2^{k/6.45}$ and impossible when $d\geq (k+1) 2^{k-1}$. In addition to the gap in~$d$, their impossibility result left open the question of whether the feasibility threshold for one-shot learning is dictated by the satisfiability threshold for bounded-degree $k$-SAT formulas. Our main contribution is to answer this question negatively. We show that one-shot learning for $k$-SAT is infeasible well below the satisfiability threshold; in fact, we obtain impossibility results for degrees $d$ as low as $k^2$ when $β$ is sufficiently large, and bootstrap this to small values of $β$ when $d$ scales exponentially with $k$, via a probabilistic construction. On the positive side, we simplify the analysis of the learning algorithm, obtaining significantly stronger bounds on $d$ in terms of $β$. For the uniform case $β\rightarrow 0$, we show that learning is possible under the condition $d\lesssim 2^{k/2}$. This is (up to constant factors) all the way to the sampling threshold -- it is known that sampling a uniformly-distributed satisfying assignment is NP-hard for $d\gtrsim 2^{k/2}$.

We study the approximability of computing the partition functions of two-state spin systems. The problem is parameterized by a 2 × 2 symmetric matrix. Previous results on this problem were restricted either to the case where the matrix has non-negative entries, or to the case where the diagonal entries are equal, i.e. Ising models. In this paper, we study the generalization to arbitrary 2 × 2 interaction matrices with real entries. We show that in some regions of the parameter space, it's #P-hard to even determine the sign of the partition function, while in other regions there are fully polynomial approximation schemes for the partition function. Our results reveal several new computational phase transitions.

Consider a k -SAT formula Φ where every variable appears at most d times, and let σ be a satisfying assignment of Φ sampled proportionally to e β m ( σ ) where m ( σ ) is the number of variables set to true and β is a real parameter. Given Φ and σ , can we learn the value of β efficiently? This problem falls into a recent line of works about single-sample (“one-shot”) learning of Markov random fields. The k -SAT setting we consider here was recently studied by Galanis, A., Kalavasis, A., Kandiros, A.V., 2024. Learning hard-constrained models with one sample. Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), 3184–3196. They showed that single-sample learning is possible when roughly d ≤ 2 k / 6.45 and impossible when d ≥ ( k + 1 ) 2 k − 1 . Crucially, for their impossibility results they used the existence of unsatisfiable instances which, aside from the gap in d , left open the question of whether the feasibility threshold for one-shot learning is dictated by the satisfiability threshold of k -SAT formulas of bounded degree. Our main contribution is to answer this question negatively. We show that one-shot learning for k -SAT is infeasible well below the satisfiability threshold; in fact, we obtain impossibility results for degrees d as low as k 2 when β is sufficiently large, and bootstrap this to small values of β when d scales exponentially with k , via a probabilistic construction. On the positive side, we simplify the analysis of the learning algorithm and obtain significantly stronger bounds on d in terms of β . In particular, for the uniform case β → 0 that has been studied extensively in the sampling literature, our analysis shows that learning is possible under the condition d ≲ 2 k / 2 . Note that this is (up to constant factors) all the way to the sampling threshold – it is known that sampling a uniformly-distributed satisfying assignment is NP-hard for d ≳ 2 k / 2 .

We study the complexity of approximating the number of answers to a small query \(\varphi\) in a large database \(\mathcal{D}\) . We establish an exhaustive classification into tractable and intractable cases if \(\varphi\) is a conjunctive query possibly including disequalities and negations: — If there is a constant bound on the arity of \(\varphi\) , and if the randomised Exponential Time Hypothesis (rETH) holds, then the problem has a fixed-parameter tractable approximation scheme (FPTRAS) if and only if the treewidth of \(\varphi\) is bounded. — If the arity is unbounded and \(\varphi\) does not have negations, then the problem has an FPTRAS if and only if the adaptive width of \(\varphi\) (a width measure strictly more general than treewidth) is bounded; the lower bound relies on the rETH as well. Additionally we show that our results cannot be strengthened to achieve a fully polynomial randomised approximation scheme (FPRAS): We observe that, unless \(\mathrm{NP}=\mathrm{RP}\) , there is no FPRAS even if the treewidth (and the adaptive width) is \(1\) . However, if there are neither disequalities nor negations, we prove the existence of an FPRAS for queries of bounded fractional hypertreewidth, strictly generalising the recently established FPRAS for conjunctive queries with bounded hypertreewidth due to Arenas, Croquevielle, Jayaram and Riveros (STOC 2021).

We study the approximability of computing the partition functions of two-state spin systems. The problem is parameterized by a 2×2 symmetric matrix. Previous results on this problem were restricted either to the case where the matrix has non-negative entries, or to the case where the diagonal entries are equal, i.e. Ising models. In this paper, we study the generalization to arbitrary 2×2 interaction matrices with real entries. We show that in some regions of the parameter space, it’s #P-hard to even determine the sign of the partition function, while in other regions there are fully polynomial approximation schemes for the partition function. Our results reveal several new computational phase transitions.

Abstract Given a class of graphs $${\mathcal {H}}$$ H , the problem $$\oplus \text {{Sub}}({\mathcal {H}})$$ ⊕ Sub ( H ) is defined as follows. The input is a graph $$H\in {\mathcal {H}}$$ H ∈ H together with an arbitrary graph G. The problem is to compute, modulo 2, the number of subgraphs of G that are isomorphic to H. The goal of this research is to determine for which classes $${\mathcal {H}}$$ H the problem $$\oplus \text {{Sub}}({\mathcal {H}})$$ ⊕ Sub ( H ) is fixed-parameter tractable (FPT), i.e., solvable in time $$f(|H|)\cdot |G|^{O(1)}$$ f ( | H | ) · | G | O ( 1 ) . Curticapean, Dell, and Husfeldt (ESA 2021) conjectured that $$\oplus \text {{Sub}}({\mathcal {H}})$$ ⊕ Sub ( H ) is FPT if and only if the class of allowed patterns $${\mathcal {H}}$$ H is matching splittable, which means that for some fixed B, every $$H \in {\mathcal {H}}$$ H ∈ H can be turned into a matching (a graph in which every vertex has degree at most 1) by removing at most B vertices. Assuming the randomised Exponential Time Hypothesis, we prove their conjecture for (I) all hereditary pattern classes $${\mathcal {H}}$$ H , and (II) all tree pattern classes, i.e., all classes $${\mathcal {H}}$$ H such that every $$H\in {\mathcal {H}}$$ H ∈ H is a tree. We also establish almost tight fine-grained upper and lower bounds for the case of hereditary patterns (I).

A graph parameter is a function f on graphs with the property that, for any pair of isomorphic graphs G 1 and G 2 , f(G 1 )=f(G 2 ). The Weisfeiler--Leman (WL) dimension of f is the minimum k such that, if G 1 and G 2 are indistinguishable by the k-dimensional WL-algorithm then f(G 1 )=f(G 2 ). The WL-dimension of f is ∞ if no such k exists. We study the WL-dimension of graph parameters characterised by the number of answers from a fixed conjunctive query to the graph. Given a conjunctive query φ, we quantify the WL-dimension of the function that maps every graph G to the number of answers of φ in G. The works of Dvorak (J. Graph Theory 2010), Dell, Grohe, and Rattan (ICALP 2018), and Neuen (ArXiv 2023) have answered this question for full conjunctive queries, which are conjunctive queries without existentially quantified variables. For such queries φ, the WL-dimension is equal to the treewidth of the Gaifman graph of φ. In this work, we give a characterisation that applies to all conjunctive queries. Given any conjunctive query φ, we prove that its WL-dimension is equal to the semantic extension width sew(φ), a novel width measure that can be thought of as a combination of the treewidth of φ and its quantified star size, an invariant introduced by Durand and Mengel (ICDT 2013) describing how the existentially quantified variables of φ are connected with the free variables. Using the recently established equivalence between the WL-algorithm and higher-order Graph Neural Networks (GNNs) due to Morris et al. (AAAI 2019), we obtain as a consequence that the function counting answers to a conjunctive query φ cannot be computed by GNNs of order smaller than sew(φ). The majority of the paper is concerned with establishing a lower bound of the WL-dimension of a query. Given any conjunctive query φ with semantic extension width k, we consider a graph F of treewidth k obtained from the Gaifman graph of φ by repeatedly cloning the vertices corresponding to existentially quantified variables. Using a modification due to Furer (ICALP 2001) of the Cai-Fürer-Immerman construction (Combinatorica 1992), we then obtain a pair of graphs χ(F) and ^χ(F) that are indistinguishable by the (k-1)-dimensional WL-algorithm since F has treewidth k. Finally, in the technical heart of the paper, we show that φ has a different number of answers in χ(F) and ^χ(F). Thus, φ can distinguish two graphs that cannot be distinguished by the (k-1)-dimensional WL-algorithm, so the WL-dimension of φ is at least k.

We study the problem of counting answers to unions of conjunctive queries (UCQs) under structural restrictions on the input query. Concretely, given a class C of UCQs, the problem #UCQ (C) provides as input a UCQ Ψ ∈ C and a database D and the problem is to compute the number of answers of Ψ in D. Chen and Mengel [PODS'16] have shown that for any recursively enumerable class C, the problem #UCQ (C) is either fixed-parameter tractable or hard for one of the parameterised complexity classes W[1] or #W[1]. However, their tractability criterion is unwieldy in the sense that, given any concrete class C of UCQs, it is not easy to determine how hard it is to count answers to queries in C. Moreover, given a single specific UCQ Ψ, it is not easy to determine how hard it is to count answers to Ψ. In this work, we address the question of finding a natural tractability criterion: The combined conjunctive query of a UCQ Ψ=φ 1 ∨ ... ∨ φ l is the conjunctive query ^ Ψ = φ_1 ∧ ... ∧ φ l . We show that under natural closure properties of C, the problem #UCQ (C) is fixed-parameter tractable if and only if the combined conjunctive queries of UCQs in C, and their contracts, have bounded treewidth. A contract of a conjunctive query is an augmented structure, taking into account how the quantified variables are connected to the free variables --- if all variables are free, then a conjunctive query is equal to its contract; in this special case the criterion for fixed-parameter tractability of #UCQ (C) thus simplifies to the combined queries having bounded treewidth. Finally, we give evidence that a closure property on C is necessary for obtaining a natural tractability criterion: We show that even for a single UCQ Ψ, the meta problem of deciding whether #UCQ (Ψ) can be solved in time O(|D| d ) is NP-hard for any fixed d ≥ 1. Moreover, we prove that a known exponential-time algorithm for solving the meta problem is optimal under assumptions from fine-grained complexity theory. As a corollary of our reduction, we also establish that approximating the Weisfeiler-Leman-Dimension of a UCQ is NP-hard.

Abstract. We study the problems of counting copies and induced copies of a small pattern graph [Formula: see text] in a large host graph [Formula: see text]. Recent work fully classified the complexity of those problems according to structural restrictions on the patterns [Formula: see text]. In this work, we address the more challenging task of analyzing the complexity for restricted patterns and restricted hosts. Specifically, we ask which families of allowed patterns and hosts imply fixed-parameter tractability, i.e., the existence of an algorithm running in time [Formula: see text] for some computable function [Formula: see text]. Our main results present exhaustive and explicit complexity classifications for families that satisfy natural closure properties. Among others, we identify the problems of counting small matchings and independent sets in subgraph-closed graph classes [Formula: see text] as our central objects of study and establish the following crisp dichotomies as consequences of the exponential time hypothesis: (1) Counting [Formula: see text]-matchings in a graph [Formula: see text] is fixed-parameter tractable if and only if [Formula: see text] is nowhere dense. (2) Counting [Formula: see text]-independent sets in a graph [Formula: see text] is fixed-parameter tractable if and only if [Formula: see text] is nowhere dense. Moreover, we obtain almost tight conditional lower bounds if [Formula: see text] is somewhere dense, i.e., not nowhere dense. These base cases of our classifications subsume a wide variety of previous results on the matching and independent set problem, such as counting [Formula: see text]-matchings in bipartite graphs (Curticapean, Marx; FOCS 14), in [Formula: see text]-colorable graphs (Roth, Wellnitz; SODA 20), and in degenerate graphs (Bressan, Roth; FOCS 21), as well as counting [Formula: see text]-independent sets in bipartite graphs (Curticapean et al.; Algorithmica 19). At the same time, our proofs are much simpler: using structural characterizations of somewhere dense graphs, we show that a colorful version of a recent breakthrough technique for analyzing pattern counting problems (Curticapean, Dell, Marx; STOC 17) applies to any subgraph-closed somewhere dense class of graphs, yielding a unified view of our current understanding of the complexity of subgraph counting.

Abstract. We give a nearly linear-time algorithm to approximately sample satisfying assignments in the random [Formula: see text]-SAT model when the density of the formula scales exponentially with [Formula: see text]. The best previously known sampling algorithm for the random [Formula: see text]-SAT model applies when the density [Formula: see text] of the formula is less than [Formula: see text] and runs in time [Formula: see text] [Galanis et al., SIAM J. Comput., 50 (2021), pp. 1701–1738]. Here [Formula: see text] is the number of variables and [Formula: see text] is the number of clauses. Our algorithm achieves a significantly faster running time of [Formula: see text] and samples satisfying assignments up to density [Formula: see text]. The main challenge in our setting is the presence of many variables with unbounded degree, which causes significant correlations within the formula and impedes the application of relevant Markov chain methods from the bounded-degree setting [Feng et al., J. ACM, 68 (2021) 40; Jain, Pham, and Vuong, On the Sampling Lovász Local Lemma for Atomic Constraint Satisfaction Problems, 2021]. Our main technical contribution is a [Formula: see text] bound of the sum of influences in the [Formula: see text]-SAT model which turns out to be robust against the presence of high-degree variables. This allows us to apply the spectral independence framework and obtain fast mixing results of a uniform-block Glauber dynamics on a carefully selected subset of the variables. The final key ingredient in our method is to take advantage of the sparsity of logarithmic-sized connected sets and the expansion properties of the random formula, and establish relevant connectivity properties of the set of satisfying assignments that enable the fast simulation of this Glauber dynamics. Our results also allow us to conclude that, with high probability, a random [Formula: see text]-CNF formula with density at most [Formula: see text] has a giant component of solutions that are connected in a graph where solutions are adjacent if they have Hamming distance [Formula: see text]. We are also able to deduce looseness results for random [Formula: see text]-CNFs in the same regime.

Spectral independence is a recently developed framework for obtaining sharp bounds on the convergence time of the classical Glauber dynamics. This new framework has yielded optimal O(n log n) sampling algorithms on bounded-degree graphs for a large class of problems throughout the so-called uniqueness regime, including, for example, the problems of sampling independent sets, matchings, and Ising-model configurations. Our main contribution is to relax the bounded-degree assumption that has so far been important in establishing and applying spectral independence. Previous methods for avoiding degree bounds rely on using L p -norms to analyse contraction on graphs with bounded connective constant (Sinclair, Srivastava, and Yin, FOCS’13). The non-linearity of L p -norms is an obstacle to applying these results to bound spectral independence. Our solution is to capture the L p -analysis recursively by amortising over the subtrees of the recurrence used to analyse contraction. Our method generalises previous analyses that applied only to bounded-degree graphs. As a main application of our techniques, we consider the random graph G (n, d/n) , where the previously known algorithms run in time n O (log d ) or applied only to large d . We refine these algorithmic bounds significantly, and develop fast nearly linear algorithms based on Glauber dynamics that apply to all constant d , throughout the uniqueness regime.

The multi-type Moran process is an evolutionary process on a connected graph G in which each vertex has one of k types and, in each step, a vertex v is chosen to reproduce its type to one of its neighbours. The probability of a vertex v being chosen for reproduction is proportional to the fitness of the type of v . So far, the literature was almost solely concerned with the 2-type Moran process in which each vertex is either healthy (type 0) or a mutant (type 1), and the main problem of interest has been the (approximate) computation of the so-called fixation probability , i.e., the probability that eventually all vertices are mutants. In this work we initiate the study of approximating fixation probabilities in the multi-type Moran process on general graphs. Our main result is an FPTRAS (fixed-parameter tractable randomised approximation scheme) for computing the fixation probability of the dominant mutation; the parameter is the number of types and their fitnesses. In the course of our studies we also provide novel upper bounds on the expected absorption time , i.e., the time that it takes the multi-type Moran process to reach a state in which each vertex has the same type.

We consider the performance of Glauber dynamics for the random cluster model with real parameter $q>1$ and temperature $β>0$. Recent work by Helmuth, Jenssen and Perkins detailed the ordered/disordered transition of the model on random $Δ$-regular graphs for all sufficiently large $q$ and obtained an efficient sampling algorithm for all temperatures $β$ using cluster expansion methods. Despite this major progress, the performance of natural Markov chains, including Glauber dynamics, is not yet well understood on the random regular graph, partly because of the non-local nature of the model (especially at low temperatures) and partly because of severe bottleneck phenomena that emerge in a window around the ordered/disordered transition. Nevertheless, it is widely conjectured that the bottleneck phenomena that impede mixing from worst-case starting configurations can be avoided by initialising the chain more judiciously. Our main result establishes this conjecture for all sufficiently large $q$ (with respect to $Δ$). Specifically, we consider the mixing time of Glauber dynamics initialised from the two extreme configurations, the all-in and all-out, and obtain a pair of fast mixing bounds which cover all temperatures $β$, including in particular the bottleneck window. Our result is inspired by the recent approach of Gheissari and Sinclair for the Ising model who obtained a similar-flavoured mixing-time bound on the random regular graph for sufficiently low temperatures. To cover all temperatures in the RC model, we refine appropriately the structural results of Helmuth, Jenssen and Perkins about the ordered/disordered transition and show spatial mixing properties "within the phase", which are then related to the evolution of the chain.

Given a class of graphs $\mathcal{H}$, the problem $\oplus\mathsf{Sub}(\mathcal{H})$ is defined as follows. The input is a graph $H\in \mathcal{H}$ together with an arbitrary graph $G$. The problem is to compute, modulo $2$, the number of subgraphs of $G$ that are isomorphic to $H$. The goal of this research is to determine for which classes $\mathcal{H}$ the problem $\oplus\mathsf{Sub}(\mathcal{H})$ is fixed-parameter tractable (FPT), i.e., solvable in time $f(|H|)\cdot |G|^{O(1)}$. Curticapean, Dell, and Husfeldt (ESA 2021) conjectured that $\oplus\mathsf{Sub}(\mathcal{H})$ is FPT if and only if the class of allowed patterns $\mathcal{H}$ is "matching splittable", which means that for some fixed $B$, every $H \in \mathcal{H}$ can be turned into a matching (a graph in which every vertex has degree at most $1$) by removing at most $B$ vertices. Assuming the randomised Exponential Time Hypothesis, we prove their conjecture for (I) all hereditary pattern classes $\mathcal{H}$, and (II) all tree pattern classes, i.e., all classes $\mathcal{H}$ such that every $H\in \mathcal{H}$ is a tree. We also establish almost tight fine-grained upper and lower bounds for the case of hereditary patterns (I).

We study the problems of counting copies and induced copies of a small pattern graph H in a large host graph G. Recent work fully classified the complexity of those problems according to structural restrictions on the patterns H. In this work, we address the more challenging task of analysing the complexity for restricted patterns and restricted hosts. Specifically we ask which families of allowed patterns and hosts imply fixed-parameter tractability, i.e., the existence of an algorithm running in time f(H)⋅|G|^O(1) for some computable function f. Our main results present exhaustive and explicit complexity classifications for families that satisfy natural closure properties. Among others, we identify the problems of counting small matchings and independent sets in subgraph-closed graph classes 𝒢 as our central objects of study and establish the following crisp dichotomies as consequences of the Exponential Time Hypothesis: - Counting k-matchings in a graph G ∈ 𝒢 is fixed-parameter tractable if and only if 𝒢 is nowhere dense. - Counting k-independent sets in a graph G ∈ 𝒢 is fixed-parameter tractable if and only if 𝒢 is nowhere dense. Moreover, we obtain almost tight conditional lower bounds if 𝒢 is somewhere dense, i.e., not nowhere dense. These base cases of our classifications subsume a wide variety of previous results on the matching and independent set problem, such as counting k-matchings in bipartite graphs (Curticapean, Marx; FOCS 14), in F-colourable graphs (Roth, Wellnitz; SODA 20), and in degenerate graphs (Bressan, Roth; FOCS 21), as well as counting k-independent sets in bipartite graphs (Curticapean et al.; Algorithmica 19). At the same time our proofs are much simpler: using structural characterisations of somewhere dense graphs, we show that a colourful version of a recent breakthrough technique for analysing pattern counting problems (Curticapean, Dell, Marx; STOC 17) applies to any subgraph-closed somewhere dense class of graphs, yielding a unified view of our current understanding of the complexity of subgraph counting.

Ranking functions such as PageRank assign numeric values (ranks) to nodes of graphs, most notably the web graph. Node rankings are an integral part of Internet search algorithms, since they can be used to order the results of queries. However, these ranking functions are famously subject to attacks by spammers, who modify the web graph in order to give their own pages more rank.

In contention resolution, multiple processors are trying to coordinate to send discrete messages through a shared channel with sharply limited communication. If two processors inadvertently send at the same time, the messages collide and are not transmitted successfully. An important case is acknowledgement-based contention resolution, in which processors cannot listen to the channel at all; all they know is whether or not their own messages have got through. This situation arises frequently in both networking and cloud computing. The most common acknowledgement-based protocols in practice are backoff protocols — variants of binary exponential backoff are used in both Ethernet and TCP/IP, and both Google Drive and AWS instruct their users to implement it to handle busy periods. In queueing models, where each processor has a queue of messages, stable backoff protocols are already known (Håstad et al., SICOMP 1996). In queue-free models, where each processor has a single message but processors arrive randomly, it is a long-standing conjecture of Aldous (IEEE Trans. Inf. Theory 1987) that no stable backoff protocols exist for any positive arrival rate of processors. Despite exciting recent results for full-sensing protocols which assume far greater listening capabilities of the processors (see e.g. Bender et al. STOC 2020 or Chen et al. PODC 2021), this foundational question remains open; here instability is only known in general when the arrival rate of processors is at least 0.42 (Goldberg et al. SICOMP 2004). We prove Aldous's conjecture for all backoff protocols outside of a tightly-constrained special case using a new domination technique to get around the main difficulty, which is the strong dependencies between messages.

The independent set polynomial is important in many areas of combinatorics, computer science, and statistical physics. For every integer Δ≥2, the Shearer threshold is the value λ⁎(Δ)=(Δ−1)Δ−1/ΔΔ. It is known that for λ<−λ⁎(Δ), there are graphs G with maximum degree Δ whose independent set polynomial, evaluated at λ, is at most 0. Also, there are no such graphs for any λ>−λ⁎(Δ). This paper is motivated by the computational problem of approximating the independent set polynomial when λ<−λ⁎(Δ). The key issue in complexity bounds for this problem is “implementation”. Informally, an implementation of a real number λ′ is a graph whose hard-core partition function, evaluated at λ, simulates a vertex-weight of λ′ in the sense that λ′ is the ratio between the contribution to the partition function from independent sets containing a certain vertex and the contribution from independent sets that do not contain that vertex. Implementations are the cornerstone of intractability results for the problem of approximately evaluating the independent set polynomial. Our main result is that, for any λ<−λ⁎(Δ), it is possible to implement a set of values that is dense over the reals. The result is tight in the sense that it is not possible to implement a set of values that is dense over the reals for any λ>λ⁎(Δ). Our result has already been used in a paper with Bezáková (STOC 2018) to show that it is #P-hard to approximate the evaluation of the independent set polynomial on graphs of degree at most Δ at any value λ<−λ⁎(Δ). In the appendix, we give an additional incomparable inapproximability result (strengthening the inapproximability bound to an exponential factor, but weakening the hardness to NP-hardness).

Spectral independence is a recently-developed framework for obtaining sharp bounds on the convergence time of the classical Glauber dynamics. This new framework has yielded optimal $O(n \log n)$ sampling algorithms on bounded-degree graphs for a large class of problems throughout the so-called uniqueness regime, including, for example, the problems of sampling independent sets, matchings, and Ising-model configurations. Our main contribution is to relax the bounded-degree assumption that has so far been important in establishing and applying spectral independence. Previous methods for avoiding degree bounds rely on using $L^p$-norms to analyse contraction on graphs with bounded connective constant (Sinclair, Srivastava, Yin; FOCS'13). The non-linearity of $L^p$-norms is an obstacle to applying these results to bound spectral independence. Our solution is to capture the $L^p$-analysis recursively by amortising over the subtrees of the recurrence used to analyse contraction. Our method generalises previous analyses that applied only to bounded-degree graphs. As a main application of our techniques, we consider the random graph $G(n,d/n)$, where the previously known algorithms run in time $n^{O(\log d)}$ or applied only to large $d$. We refine these algorithmic bounds significantly, and develop fast $n^{1+o(1)}$ algorithms based on Glauber dynamics that apply to all $d$, throughout the uniqueness regime.

We study the performance of Markov chains for the $q$-state ferromagnetic Potts model on random regular graphs. It is conjectured that their performance is dictated by metastability phenomena, i.e., the presence of "phases" (clusters) in the sample space where Markov chains with local update rules, such as the Glauber dynamics, are bound to take exponential time to escape. The phases that are believed to drive these metastability phenomena in the case of the Potts model emerge as local, rather than global, maxima of the so-called Bethe functional, and previous approaches of analysing these phases based on optimisation arguments fall short of the task. Our first contribution is to detail the emergence of the metastable phases for the $q$-state Potts model on the $d$-regular random graph for all integers $q,d\geq 3$, and establish that for an interval of temperatures, which is delineated by the uniqueness and a broadcasting threshold on the $d$-regular tree, the two phases coexist. The proofs are based on a conceptual connection between spatial properties and the structure of the Potts distribution on the random regular graph, rather than complicated moment calculations. Based on this new structural understanding of the model, we obtain various algorithmic consequences. We first complement recent fast mixing results for Glauber dynamics by Blanca and Gheissari below the uniqueness threshold, showing an exponential lower bound on the mixing time above the uniqueness threshold. Then, we obtain tight results even for the non-local Swendsen-Wang chain, where we establish slow mixing/metastability for the whole interval of temperatures where the chain is conjectured to mix slowly on the random regular graph. The key is to bound the conductance of the chains using a random graph "planting" argument combined with delicate bounds on random-graph percolation.

We study the complexity of approximating the number of answers to a small query φ in a large database D. We establish an exhaustive classification into tractable and intractable cases if φ is a conjunctive query possibly including disequalities and negations: - If there is a constant bound on the arity of φ, and if the randomised Exponential Time Hypothesis (rETH) holds, then the problem has a fixed-parameter tractable approximation scheme (FPTRAS) if and only if the treewidth of φ is bounded. - If the arity is unbounded and φ does not have negations, then the problem has an FPTRAS if and only if the adaptive width of φ (a width measure strictly more general than treewidth) is bounded; the lower bound relies on the rETH as well. Additionally we show that our results cannot be strengthened to achieve a fully polynomial randomised approximation scheme (FPRAS): We observe that, unless NP=RP, there is no FPRAS even if the treewidth (and the adaptive width) is 1. However, if there are neither disequalities nor negations, we prove the existence of an FPRAS for queries of bounded fractional hypertreewidth, strictly generalising the recently established FPRAS for conjunctive queries with bounded hypertreewidth due to Arenas, Croquevielle, Jayaram and Riveros (STOC 2021).

Abstract We study the complexity of approximating the partition function of the q-state Potts model and the closely related Tutte polynomial for complex values of the underlying parameters. Apart from the classical connections with quantum computing and phase transitions in statistical physics, recent work in approximate counting has shown that the behaviour in the complex plane, and more precisely the location of zeros, is strongly connected with the complexity of the approximation problem, even for positive real-valued parameters. Previous work in the complex plane by Goldberg and Guo focused on q = 2, which corresponds to the case of the Ising model; for q > 2, the behaviour in the complex plane is not as well understood and most work applies only to the real-valued Tutte plane. Our main result is a complete classification of the complexity of the approximation problems for all non-real values of the parameters, by establishing #P-hardness results that apply even when restricted to planar graphs. Our techniques apply to all q $$\geq$$ ≥ 2 and further complement/refine previous results both for the Ising model and the Tutte plane, answering in particular a question raised by Bordewich, Freedman, Lovász and Welsh in the context of quantum computations.

The polymer model framework is a classical tool from statistical mechanics that has recently been used to obtain approximation algorithms for spin systems on classes of bounded-degree graphs; examples include the ferromagnetic Potts model on expanders and on the grid. One of the key ingredients in the analysis of polymer models is controlling the growth rate of the number of polymers, which has been typically achieved so far by invoking the bounded-degree assumption. Nevertheless, this assumption is often restrictive and obstructs the applicability of the method to more general graphs. For example, sparse random graphs typically have bounded average degree and good expansion properties, but they include vertices with unbounded degree, and therefore are excluded from the current polymer-model framework. We develop a less restrictive framework for polymer models that relaxes the standard bounded-degree assumption, by reworking the relevant polymer models from the edge perspective. The edge perspective allows us to bound the growth rate of the number of polymers in terms of the total degree of polymers, which in turn can be related more easily to the expansion properties of the underlying graph. To apply our methods, we consider random graphs with unbounded degrees from a fixed degree sequence (with minimum degree at least 3) and obtain approximation algorithms for the ferromagnetic Potts model, which is a standard benchmark for polymer models. Our techniques also extend to more general spin systems.

We study the complexity of approximating the partition function $Z_{\mathrm{Ising}}(G; \beta)$ of the Ising model in terms of the relation between the edge interaction $\beta$ and a parameter $\Delta$ which is an upper bound on the maximum degree of the input graph $G$. Following recent trends in both statistical physics and algorithmic research, we allow the edge interaction $\beta$ to be any complex number. Many recent partition function results focus on complex parameters, both because of physical relevance and because of the key role of the complex case in delineating the tractability/intractability phase transition of the approximation problem. In this work we establish both new tractability results and new intractability results. Our tractability results show that $Z_{\mathrm{Ising}}(-; \beta)$ has an FPTAS when $\lvert \beta - 1 \rvert / \lvert \beta + 1 \rvert < \tan(\pi / (4 \Delta - 4))$. The core of the proof is showing that there are no inputs $G$ that make the partition function $0$ when $\beta$ is in this range. Our result significantly extends the known zero-free region of the Ising model (and hence the known approximation results). Our intractability results show that it is $\mathrm{\#P}$-hard to multiplicatively approximate the norm and to additively approximate the argument of $Z_{\mathrm{Ising}}(-; \beta)$ when $\beta \in \mathbb{C}$ is an algebraic number such that $\beta \not \in \mathbb{R} \cup \{i, -i\}$ and $\lvert \beta - 1\rvert / \lvert \beta + 1 \rvert > 1 / \sqrt{\Delta - 1}$. These are the first results to show intractability of approximating $Z_{\mathrm{Ising}}(-, \beta)$ on bounded degree graphs with complex $\beta$. Moreover, we demonstrate situations in which zeros of the partition function imply hardness of approximation in the Ising model.

A homomorphism from a graph G to a graph H is a function from the vertices of G to the vertices of H that preserves the edges of G in the sense that every edge of G is mapped to an edge of H. By changing the target graph H, we can capture interesting structures in G. For example, homomorphisms from G to a k-clique H correspond to the proper k-colourings of G. There has been a lot of algorithmic work on the problem of (approximately) counting homomorphisms. The goal is to figure out for which graphs H the problem of approximately counting homomorphisms to H is algorithmically feasible. This talk will survey what is known. Despite much work, there are still plenty of open problems. We will discuss the problem of approximately counting list homomorphisms (where the input specifies, for each vertex of G, the list of vertices of H to which it can be mapped). Because the lists add extra expressibility, it is easier to prove that counting homomorphisms to a particular graph H is intractable. In fact, we have a full trichotomy (joint work with Galanis and Jerrum, 2017). Here, the complexity of homomorphism-counting is related to certain hereditary graph classes. The trichotomy will be explained in the talk - no prior knowledge of the area will be assumed. In more recent work, with Focke and Živn{ý}, we have investigated the complexity of counting retractions to H - this problem falls between homomorphism-counting and list-homomorphism counting. Here we have only a partial classification, which applies to all square-free graphs H. So again, there are plenty of open problems.

We study the problem of computing the parity of the number of homomorphisms from an input graph G to a fixed graph H. Faben and Jerrum [ToC'15] introduced an explicit criterion on the graph H and conjectured that, if satisfied, the problem is solvable in polynomial time and, otherwise, the problem is complete for the complexity class ⊕P of parity problems. We verify their conjecture for all graphs H that exclude the complete graph on 4 vertices as a minor. Further, we rule out the existence of a subexponential-time algorithm for the ⊕P-complete cases, assuming the randomised Exponential Time Hypothesis. Our proofs introduce a novel method of deriving hardness from globally defined substructures of the fixed graph H. Using this, we subsume all prior progress towards resolving the conjecture (Faben and Jerrum [ToC'15]; Göbel, Goldberg and Richerby [ToCT'14,'16]). As special cases, our machinery also yields a proof of the conjecture for graphs with maximum degree at most 3, as well as a full classification for the problem of counting list homomorphisms, modulo 2. A full version of our paper, containing all proofs, is available at https://arxiv.org/abs/2006.16632v2. Here we number key lemmas to match the numbering in the full version.

We give the first efficient algorithm to approximately count the number of solutions in the random $k$-SAT model when the density of the formula scales exponentially with $k$. The best previous counting algorithm for the permissive version of the model was due to Montanari and Shah and was based on the correlation decay method, which works up to densities $(1+o_k(1))\frac{2\log k}{k}$, the Gibbs uniqueness threshold for the model. Instead, our algorithm harnesses a recent technique by Moitra to work for random formulas with much higher densities. The main challenge in our setting is to account for the presence of high-degree variables whose marginal distributions are hard to control and which cause significant correlations within the formula.

We study the problem of computing the parity of the number of homomorphisms from an input graph $G$ to a fixed graph $H$. Faben and Jerrum [ Theory Comput., 11 (2015), pp. 35--57] introduced an explicit criterion on the graph $H$ and conjectured that, if satisfied, the problem is solvable in polynomial time and, otherwise, the problem is complete for the complexity class $\oplus{P}$ of parity problems. We verify their conjecture for all graphs $H$ that exclude the complete graph on four vertices as a minor. Further, we rule out the existence of a subexponential-time algorithm for the $\oplus{P}$-complete cases, assuming the randomized exponential time hypothesis. Our proofs introduce a novel method of deriving hardness from globally defined substructures of the fixed graph $H$. Using this, we subsume all prior progress toward resolving the conjecture (Faben and Jerrum [ Theory Comput., 11 (2015), pp. 35--57]; Göbel, Goldberg, and Richerby [ ACM Trans. Comput. Theory, 6 (2014), 17; ACM Trans. Comput. Theory, 8 (2016), 12]). As special cases, our machinery also yields a proof of the conjecture for graphs with maximum degree at most 3, as well as a full classification for the problem of counting list homomorphisms, modulo 2.

A retraction is a homomorphism from a graph G to an induced subgraph H of G that is the identity on H . In a long line of research, retractions have been studied under various algorithmic settings. Recently, the problem of approximately counting retractions was considered. We give a complete trichotomy for the complexity of approximately counting retractions to all square-free graphs (graphs that do not contain a cycle of length 4). It turns out there is a rich and interesting class of graphs for which this problem is complete in the class #BIS. As retractions generalise homomorphisms, our easiness results extend to the important problem of approximately counting homomorphisms. By giving new #BIS-easiness results, we now settle the complexity of approximately counting homomorphisms for a whole class of non-trivial graphs that were previously unresolved.

We give the first efficient algorithm to approximately count the number of solutions in the random $k$-SAT model when the density of the formula scales exponentially with $k$. The best previous counting algorithm for the permissive version of the model was due to Montanari and Shah and was based on the correlation decay method, which works up to densities $(1+o_k(1))\frac{2\log k}{k}$, the Gibbs uniqueness threshold for the model. Instead, our algorithm harnesses a recent technique by Moitra to work for random formulas. The main challenge in our setting is to account for the presence of high-degree variables whose marginal distributions are hard to control and which cause significant correlations within the formula.

We study the complexity of approximating the partition function of the q-state Potts model and the closely related Tutte polynomial for complex values of the underlying parameters. Apart from the classical connections with quantum computing and phase transitions in statistical physics, recent work in approximate counting has shown that the behaviour in the complex plane, and more precisely the location of zeros, is strongly connected with the complexity of the approximation problem, even for positive real-valued parameters. Previous work in the complex plane by Goldberg and Guo focused on q = 2, which corresponds to the case of the Ising model; for q > 2, the behaviour in the complex plane is not as well understood and most work applies only to the real-valued Tutte plane. Our main result is a complete classification of the complexity of the approximation problems for all non-real values of the parameters, by establishing #P-hardness results that apply even when restricted to planar graphs. Our techniques apply to all q ≥ 2 and further complement/refine previous results both for the Ising model and the Tutte plane, answering in particular a question raised by Bordewich, Freedman, Lovász and Welsh in the context of quantum computations.

We study the complexity of approximating the value of the independent set polynomial $Z_G(\lambda)$ of a graph $G$ with maximum degree $\Delta$ when the activity $\lambda$ is a complex number. When $\lambda$ is real, the complexity picture is well understood, and is captured by two real-valued thresholds $\lambda^*$ and $\lambda_c$, which depend on $\Delta$ and satisfy $0<\lambda^*<\lambda_c$. It is known that if $\lambda$ is a real number in the interval $(-\lambda^*,\lambda_c)$ then there is a fully polynomial time approximation scheme (FPTAS) for approximating $Z_G(\lambda)$ on graphs $G$ with maximum degree at most $\Delta$. On the other hand, if $\lambda$ is a real number outside of the (closed) interval, then approximation is NP-hard. The key to establishing this picture was the interpretation of the thresholds $\lambda^*$ and $\lambda_c$ on the $\Delta$-regular tree. The “occupation ratio” of a $\Delta$-regular tree $T$ is the contribution to $Z_T(\lambda)$ from independent sets containing the root of the tree, divided by $Z_T(\lambda)$ itself. This occupation ratio converges to a limit, as the height of the tree grows, if and only if $\lambda\in [-\lambda^*,\lambda_c]$. Unsurprisingly, the case where $\lambda$ is complex is more challenging. It is known that there is an FPTAS when $\lambda$ is a complex number with norm at most $\lambda^*$ and also when $\lambda$ is in a small strip surrounding the real interval $[0,\lambda_c)$. However, neither of these results is believed to fully capture the truth about when approximation is possible. Peters and Regts identified the complex values of $\lambda$ for which the occupation ratio of the $\Delta$-regular tree converges. These values carve a cardioid-shaped region $\Lambda_\Delta$ in the complex plane, whose boundary includes the critical points $-\lambda^*$ and $\lambda_c$. Motivated by the picture in the real case, they asked whether $\Lambda_\Delta$ marks the true approximability threshold for general complex values $\lambda$. Our main result shows that for every $\lambda$ outside of $\Lambda_\Delta$, the problem of approximating $Z_G(\lambda)$ on graphs $G$ with maximum degree at most $\Delta$ is indeed NP-hard. In fact, when $\lambda$ is outside of $\Lambda_\Delta$ and is not a positive real number, we give the stronger result that approximating $Z_G(\lambda)$ is actually \#P-hard. Further, on the negative real axis, when $\lambda < - \lambda^*$, we show that it is \#P-hard to even decide whether $Z_G(\lambda)>0$, resolving in the affirmative a conjecture of Harvey, Srivastava, and Vondrák. Our proof techniques are based around tools from complex analysis---specifically the study of iterative multivariate rational maps.

We consider the problem of sampling from the Potts model on random regular graphs. It is conjectured that sampling is possible when the temperature of the model is in the so-called uniqueness regime of the regular tree, but positive algorithmic results have been for the most part elusive. In this paper, for all integers $q\geq 3$ and $\Delta\geq 3$, we develop algorithms that produce samples within error $o(1)$ from the $q$-state Potts model on random $\Delta$-regular graphs, whenever the temperature is in uniqueness, for both the ferromagnetic and antiferromagnetic cases. The algorithm for the antiferromagnetic Potts model is based on iteratively adding the edges of the graph and resampling a bichromatic class that contains the endpoints of the newly added edge. Key to the algorithm is how to perform the resampling step efficiently since bichromatic classes can potentially induce linear-sized components. To this end, we exploit the tree uniqueness to show that the average growth of bichromatic components is typically small, which allows us to use correlation decay algorithms for the resampling step. While the precise uniqueness threshold on the tree is not known for general values of $q$ and $\Delta$ in the antiferromagnetic case, our algorithm works throughout uniqueness regardless of its value. In the case of the ferromagnetic Potts model, we are able to simplify the algorithm significantly by utilizing the random-cluster representation of the model. In particular, we demonstrate that a percolation-type algorithm succeeds in sampling from the random-cluster model with parameters $p,q$ on random $\Delta$-regular graphs for all values of $q\geq 1$ and $p

We construct a theory of holant clones to capture the notion of expressibility in the holant framework. Their role is analogous to the role played by functional clones in the study of weighted counting Constraint Satisfaction Problems. We explore the landscape of conservative holant clones and determine the situations in which a set F of functions is “universal in the conservative case,” which means that all functions are contained in the holant clone generated by F together with all unary functions. When F is not universal in the conservative case, we give concise generating sets for the clone. We demonstrate the usefulness of the holant clone theory by using it to give a complete complexity-theory classification for the problem of approximating the solution to conservative holant problems. We show that approximation is intractable exactly when F is universal in the conservative case.

We study the mixing time of random walks on small-world networks modelled as follows: starting with the 2-dimensional periodic grid, each pair of vertices {u,v} with distance d> 1 is added as a “long-range” edge with probability proportional to d -r , where r≥ 0 is a parameter of the model. Kleinberg [33{ studied a close variant of this network model and proved that the (decentralised) routing time is O((log n ) 2 ) when r =2 and n Ω (1) when r≠ 2. Here, we prove that the random walk also undergoes a phase transition at r=2 , but in this case, the phase transition is of a different form. We establish that the mixing time is ϴ (log n) for r< 2, O((log n ) 4 ) for r =2, and n Ω (1) for r> 2.

For spin systems, such as the hard-core model on independent sets weighted by fugacity lambda>0, efficient algorithms for the associated approximate counting/sampling problems typically apply in the high-temperature region, corresponding to low fugacity. Recent work of Jenssen, Keevash and Perkins (2019) yields an FPTAS for approximating the partition function (and an efficient sampling algorithm) on bounded-degree (bipartite) expander graphs for the hard-core model at sufficiently high fugacity, and also the ferromagnetic Potts model at sufficiently low temperatures. Their method is based on using the cluster expansion to obtain a complex zero-free region for the partition function of a polymer model, and then approximating this partition function using the polynomial interpolation method of Barvinok. We present a simple discrete-time Markov chain for abstract polymer models, and present an elementary proof of rapid mixing of this new chain under sufficient decay of the polymer weights. Applying these general polymer results to the hard-core and ferromagnetic Potts models on bounded-degree (bipartite) expander graphs yields fast algorithms with running time O(n log n) for the Potts model and O(n^2 log n) for the hard-core model, in contrast to typical running times of n^{O(log Delta)} for algorithms based on Barvinok’s polynomial interpolation method on graphs of maximum degree Delta. In addition, our approach via our polymer model Markov chain is conceptually simpler as it circumvents the zero-free analysis and the generalization to complex parameters. Finally, we combine our results for the hard-core and ferromagnetic Potts models with standard Markov chain comparison tools to obtain polynomial mixing time for the usual spin system Glauber dynamics restricted to even and odd or "red" dominant portions of the respective state spaces.

Let G be a graph that contains an induced subgraph H. A retraction from G to H is a homomorphism from G to H that is the identity function on H. Retractions are very well-studied: Given H, the complexity of deciding whether there is a retraction from an input graph G to H is completely classified, in the sense that it is known for which H this problem is tractable (assuming P ≠ NP). Similarly, the complexity of (exactly) counting retractions from G to H is classified (assuming FP ≠ #P). However, almost nothing is known about approximately counting retractions. Our first contribution is to give a complete trichotomy for approximately counting retractions to trees. The result is as follows: (1) Approximately counting retractions to a tree H is in FP if H is a star, a single looped vertex, or an edge with two loops. (2) Otherwise, if H is an irreflexive caterpillar or a partially bristled reflexive path, then approximately counting retractions to H is equivalent to approximately counting the independent sets of a bipartite graph — a problem which is complete in the approximate counting complexity class RHπ1. (3) Finally, if none of these hold, then approximately counting retractions to H is #P-complete under approximation-preserving reductions. Our second contribution is to locate the retraction counting problem in the complexity landscape of related approximate counting problems. Interestingly, our results are in contrast to the situation in the exact counting context. We show that the problem of approximately counting retractions is separated both from the problem of approximately counting homomorphisms and from the problem of approximately counting list homomorphisms — whereas for exact counting all three of these problems are interreducible. We also show that the number of retractions is at least as hard to approximate as both the number of surjective homomorphisms and the number of compactions. In contrast, exactly counting compactions is the hardest of these problems. The full version containing detailed proofs is available at https://arxiv.org/abs/1807.00590v1 (version from 2 July 2018). The theorem numbering here matches the full version.

This paper is an extended abstract of my STACS 2019 talk "Computational Complexity and Partition Functions".

The problem of (approximately) counting the independent sets of a bipartite graph (#BIS) is the canonical approximate counting problem that is complete in the intermediate complexity class $$\mathsf {\#RH}\Pi _1$$ . It is believed that #BIS does not have an efficient approximation algorithm but also that it is not NP-hard. We study the robustness of the intermediate complexity of #BIS by considering variants of the problem parameterised by the size of the independent set. We map the complexity landscape for three problems, with respect to exact computation and approximation and with respect to conventional and parameterised complexity. The three problems are counting independent sets of a given size, counting independent sets with a given number of vertices in one vertex class and counting maximum independent sets amongst those with a given number of vertices in one vertex class. Among other things, we show that all of these problems are NP-hard to approximate within any polynomial ratio. (This is surprising because the corresponding problems without the size parameter are complete in $$\mathsf {\#RH}\Pi _1$$ , and hence are not believed to be NP-hard.) We also show that the first problem is #W[1]-hard to solve exactly but admits an FPTRAS, whereas the other two are W[1]-hard to approximate even within any polynomial ratio. Finally, we show that, when restricted to graphs of bounded degree, all three problems have efficient exact fixed-parameter algorithms.

Approximate counting via correlation decay is the core algorithmic technique used in the sharp delineation of the computational phase transition that arises in the approximation of the partition function of antiferromagnetic 2-spin models. Previous analyses of correlation-decay algorithms implicitly depended on the occurrence of strong spatial mixing. This, roughly, means that one uses worst-case analysis of the recursive procedure that creates the subinstances. In this paper, we develop a new analysis method that is more refined than the worst-case analysis. We take the shape of instances in the computation tree into consideration and we amortize against certain “bad” instances that are created as the recursion proceeds. This enables us to show correlation decay and to obtain a fully polynomial-time approximation scheme (FPTAS) even when strong spatial mixing fails. We apply our technique to the problem of approximately counting independent sets in hypergraphs with degree upper bound $\Delta$ and with a lower bound $k$ on the arity of hyperedges. Liu and Lin gave an FPTAS for $k\geq2$ and $\Delta\leq5$ (lack of strong spatial mixing was the obstacle preventing this algorithm from being generalized to $\Delta=6$). Our technique gives a tight result for $\Delta=6$, showing that there is an FPTAS for $k\geq3$ and $\Delta\leq6$. The best previously known approximation scheme for $\Delta=6$ is the Markov-chain simulation based fully polynomial-time randomized approximation scheme (FPRAS) of Bordewich, Dyer, and Karpinski, which only works for $k\geq8$. Our technique also applies for larger values of $k$, giving an FPTAS for $k\geq\Delta$. This bound is not substantially stronger than existing randomized results in the literature. Nevertheless, it gives the first deterministic approximation scheme in this regime. Moreover, unlike existing results, it leads to an FPTAS for counting dominating sets in regular graphs with sufficiently large degree. We further demonstrate that in the hypergraph independent set model, approximating the partition function is NP-hard even within the uniqueness regime. Also, approximately counting dominating sets of bounded-degree graphs (without the regularity restriction) is NP-hard.

A homomorphism from a graph $G$ to a graph $H$ is a function from the vertices of $G$ to the vertices of $H$ that preserves edges. A homomorphism is surjective if it uses all of the vertices of $H$, and it is a compaction if it uses all of the vertices of $H$ and all of the nonloop edges of $H$. Hell and Nešetřil gave a complete characterization of the complexity of deciding whether there is a homomorphism from an input graph $G$ to a fixed graph $H$. A complete characterization is not known for surjective homomorphisms or for compactions, though there are many interesting results. Dyer and Greenhill gave a complete characterization of the complexity of counting homomorphisms from an input graph $G$ to a fixed graph $H$. In this paper, we give a complete characterization of the complexity of counting surjective homomorphisms from an input graph $G$ to a fixed graph $H$, and we also give a complete characterization of the complexity of counting compactions from an input graph $G$ to a fixed graph $H$. In an addendum we use our characterizations to point out a dichotomy for the complexity of the respective approximate counting problems (in the connected case).

We consider the problem of sampling from the Potts model on random regular graphs. It is conjectured that sampling is possible when the temperature of the model is in the uniqueness regime of the regular tree, but positive algorithmic results have been for the most part elusive. In this paper, for all integers $q\geq 3$ and $Δ\geq 3$, we develop algorithms that produce samples within error $o(1)$ from the $q$-state Potts model on random $Δ$-regular graphs, whenever the temperature is in uniqueness, for both the ferromagnetic and antiferromagnetic cases. The algorithm for the antiferromagnetic Potts model is based on iteratively adding the edges of the graph and resampling a bichromatic class that contains the endpoints of the newly added edge. Key to the algorithm is how to perform the resampling step efficiently since bichromatic classes may induce linear-sized components. To this end, we exploit the tree uniqueness to show that the average growth of bichromatic components is typically small, which allows us to use correlation decay algorithms for the resampling step. While the precise uniqueness threshold on the tree is not known for general values of $q$ and $Δ$ in the antiferromagnetic case, our algorithm works throughout uniqueness regardless of its value. In the case of the ferromagnetic Potts model, we simplify the algorithm significantly by utilising the random-cluster representation of the model. In particular, we show that a percolation-type algorithm succeeds in sampling from the random-cluster model with parameters $p,q$ on random $Δ$-regular graphs for all values of $q\geq 1$ and $p

A homomorphism from a graph G to a graph H is a function from the vertices of G to the vertices of H that preserves edges. A homomorphism is surjective if it uses all of the vertices of H and it is a compaction if it uses all of the vertices of H and all of the non-loop edges of H. Hell and Nešetřil gave a complete characterisation of the complexity of deciding whether there is a homomorphism from an input graph G to a fixed graph H. A complete characterisation is not known for surjective homomorphisms or for compactions, though there are many interesting results. Dyer and Greenhill gave a complete characterisation of the complexity of counting homomorphisms from an input graph G to a fixed graph H. In this paper, we give a complete characterisation of the complexity of counting surjective homomorphisms from an input graph G to a fixed graph H and we also give a complete characterisation of the complexity of counting compactions from an input graph G to a fixed graph H.

We study the problem of approximately evaluating the independent set polynomial of bounded-degree graphs at a point lambda or, equivalently, the problem of approximating the partition function of the hard-core model with activity lambda on graphs G of max degree D. For lambda>0, breakthrough results of Weitz and Sly established a computational transition from easy to hard at lambda_c(D)=(D-1)^(D-1)/(D-2)^D, which coincides with the tree uniqueness phase transition from statistical physics. For lambda<0, the evaluation of the independent set polynomial is connected to the conditions of the Lovasz Local Lemma. Shearer identified the threshold lambda*(D)=(D-1)^(D-1)/D^D as the maximum value p such that every family of events with failure probability at most p and whose dependency graph has max degree D has nonempty intersection. Very recently, Patel and Regts, and Harvey et al. have independently designed FPTASes for approximating the partition function whenever |lambda| =3, for all lambda<-lambda*(D), it is NP-hard to approximate the partition function on graphs of maximum degree D, even within an exponential factor. Thus, our result, combined with the FPTASes for lambda>-lambda*(D), establishes a phase transition for negative activities. In fact, we now have the following picture for the problem of approximating the partition function with activity lambda on graphs G of max degree D. 1. For -lambda*(D) lambda_c(D), the problem is NP-hard. Rather than the tree uniqueness threshold of the positive case, the phase transition for negative activities corresponds to the existence of zeros for the partition function of the tree below -lambda*(D).

We study the complexity of approximate counting Constraint Satisfaction Problems (#CSPs) in a bounded degree setting. Specifically, given a Boolean constraint language Gamma and a degree bound Delta, we study the complexity of #CSP_Delta(Gamma), which is the problem of counting satisfying assignments to CSP instances with constraints from Gamma and whose variables can appear at most Delta times. Our main result shows that: (i) if every function in Gamma is affine, then #CSP_Delta(Gamma) is in FP for all Delta, (ii) otherwise, if every function in Gamma is in a class called IM_2, then for all sufficiently large Delta, #CSP_Delta(Gamma) is equivalent under approximation-preserving (AP) reductions to the counting problem #BIS (the problem of counting independent sets in bipartite graphs) (iii) otherwise, for all sufficiently large Delta, it is NP-hard to approximate the number of satisfying assignments of an instance of #CSP_Delta(Gamma), even within an exponential factor. Our result extends previous results, which apply only in the so-called "conservative" case.

The Moran process, as studied by Lieberman, Hauert, and Nowak, is a randomised algorithm modelling the spread of genetic mutations in populations. The algorithm runs on an underlying graph where individuals correspond to vertices. Initially, one vertex (chosen uniformly at random) possesses a mutation, with fitness r > 1. All other individuals have fitness 1. During each step of the algorithm, an individual is chosen with probability proportional to its fitness, and its state (mutant or nonmutant) is passed on to an out-neighbour which is chosen uniformly at random. If the underlying graph is strongly connected, then the algorithm will eventually reach fixation , in which all individuals are mutants, or extinction , in which no individuals are mutants. An infinite family of directed graphs is said to be strongly amplifying if, for every r > 1, the extinction probability tends to 0 as the number of vertices increases. A formal definition is provided in the article. Strong amplification is a rather surprising property—it means that in such graphs, the fixation probability of a uniformly placed initial mutant tends to 1 even though the initial mutant only has a fixed selective advantage of r > 1 (independently of n ). The name “strongly amplifying” comes from the fact that this selective advantage is “amplified.” Strong amplifiers have received quite a bit of attention, and Lieberman et al. proposed two potentially strongly amplifying families—superstars and metafunnels. Heuristic arguments have been published, arguing that there are infinite families of superstars that are strongly amplifying. The same has been claimed for metafunnels. In this article, we give the first rigorous proof that there is an infinite family of directed graphs that is strongly amplifying. We call the graphs in the family “megastars.” When the algorithm is run on an n -vertex graph in this family, starting with a uniformly chosen mutant, the extinction probability is roughly n − 1/2 (up to logarithmic factors). We prove that all infinite families of superstars and metafunnels have larger extinction probabilities (as a function of n ). Finally, we prove that our analysis of megastars is fairly tight—there is no infinite family of megastars such that the Moran algorithm gives a smaller extinction probability (up to logarithmic factors). Also, we provide a counterexample which clarifies the literature concerning the isothermal theorem of Lieberman et al.

We examine the computational complexity of approximately counting the list H-colourings of a graph. We discover a natural graph-theoretic trichotomy based on the structure of the graph H. If H is an irreflexive bipartite graph or a reflexive complete graph then counting list H-colourings is trivially in polynomial time. Otherwise, if H is an irreflexive bipartite permutation graph or a reflexive proper interval graph then approximately counting list H-colourings is equivalent to #BIS, the problem of approximately counting independent sets in a bipartite graph. This is a well-studied problem which is believed to be of intermediate complexity - it is believed that it does not have an FPRAS, but that it is not as difficult as approximating the most difficult counting problems in #P. For every other graph H, approximately counting list H-colourings is complete for #P with respect to approximation-preserving reductions (so there is no FPRAS unless NP = RP). Two pleasing features of the trichotomy are (i) it has a natural formulation in terms of hereditary graph classes, and (ii) the proof is largely self-contained and does not require any universal algebra (unlike similar dichotomies in the weighted case). We are able to extend the hardness results to the bounded-degree setting, showing that all hardness results apply to input graphs with maximum degree at most 6.

One of the most important recent developments in the complexity of approximate counting is the classification of the complexity of approximating the partition functions of antiferromagnetic 2-spin systems on bounded-degree graphs. This classification is based on a beautiful connection to the so-called uniqueness phase transition from statistical physics on the infinite Δ-regular tree. Our objective is to study the impact of this classification on unweighted 2-spin models on k-uniform hypergraphs. As has already been indicated by Yin and Zhao, the connection between the uniqueness phase transition and the complexity of approximate counting breaks down in the hypergraph setting. Nevertheless, we show that for every non-trivial symmetric k-ary Boolean function f there exists a degree bound Δ0 so that for all Δ ≥ Δ0 the following problem is NP-hard: given a k-uniform hypergraph with maximum degree at most Δ, approximate the partition function of the hypergraph 2-spin model associated with f. It is NP-hard to approximate this partition function even within an exponential factor. By contrast, if f is a trivial symmetric Boolean function (e.g., any function f that is excluded from our result), then the partition function of the corresponding hypergraph 2-spin model can be computed exactly in polynomial time. The full version of this paper is available at arxiv.org/abs/1505.06146. For the convenience of the reader, we have also included in the theorem statements the number of the respective theorem in the full version.

We consider the problem of counting $H$-colorings from an input graph $G$ to a target graph $H$. We show that if $H$ is any fixed graph without trivial components, then the problem is as hard as the well-known problem $\#\mathrm{BIS}$, which is the problem of (approximately) counting independent sets in a bipartite graph. $\#\mathrm{BIS}$ is a complete problem in an important complexity class for approximate counting, and is believed not to have a fully polynomial randomized approximation scheme (FPRAS). If this is so, then our result shows that for every graph $H$ without trivial components, the $H$-coloring counting problem has no FPRAS. This problem was studied a decade ago by Goldberg, Kelk, and Paterson. They were able to show that approximately sampling $H$-colorings is $\#\mathrm{BIS}$-hard, but it was not known how to get the result for approximate counting. Our solution builds on nonconstructive ideas using the work of Lovász.

No abstract available.

Given a symmetric $D\times D$ matrix $M$ over \0,1,*\, a list $M$-partition of a graph $G$ is a partition of the vertices of $G$ into $D$ parts which are associated with the rows of $M$. The part of each vertex is chosen from a given list in such a way that no edge of $G$ is mapped to a 0 in $M$ and no nonedge of $G$ is mapped to a 1 in $M$. Many important graph-theoretic structures can be represented as list $M$-partitions including graph colorings, split graphs, and homogeneous sets and pairs, which arise in the proofs of the weak and strong perfect graph conjectures. Thus, there has been quite a bit of work on determining for which matrices $M$ computations involving list $M$-partitions are tractable. This paper focuses on the problem of counting list $M$-partitions, given a graph $G$ and given a list for each vertex of $G$. We identify a certain set of “tractable” matrices $M$. We give an algorithm that counts list $M$-partitions in polynomial time for every (fixed) matrix $M$ in this set. The algorithm relies on data structures such as sparse-dense partitions and subcube decompositions to reduce each problem instance to a sequence of problem instances in which the lists have a certain useful structure that restricts access to portions of $M$ in which the interactions of 0s and 1s are controlled. We show how to solve the resulting restricted instances by converting them into particular counting constraint satisfaction problems (${\#CSP}$s), which we show how to solve using a constraint satisfaction technique known as arc-consistency. For every matrix $M$ for which our algorithm fails, we show that the problem of counting list $M$-partitions is ${\#P}$-complete. Furthermore, we give an explicit characterization of the dichotomy theorem: counting list $M$-partitions is tractable (in ${FP}$) if the matrix $M$ has a structure called a derectangularizing sequence. If $M$ has no derectangularizing sequence, we show that counting list $M$-partitions is ${\#P}$-hard. We show that the metaproblem of determining whether a given matrix has a derectangularizing sequence is ${NP}$-complete. Finally, we show that list $M$-partitions can be used to encode cardinality restrictions in $M$-partitions problems, and we use this to give a polynomial-time algorithm for counting homogeneous pairs in graphs.

Counting independent sets on bipartite graphs (#BIS) is considered a canonical counting problem of intermediate approximation complexity. It is conjectured that #BIS neither has an FPRAS nor is as hard as #SAT to approximate. We study #BIS in the general framework of two-state spin systems in bipartite graphs. Such a system is parameterized by three numbers (beta,gamma,lambda), where beta (respectively gamma) represents the weight of an edge (or "interaction strength") whose endpoints are of the same 0 (respectively 1) spin, and lambda is the weight of a 1 vertex, also known as an "external field". By convention, the edge weight with unequal 0/1 end points and the vertex weight with spin 0 are both normalized to 1. The partition function of the special case beta=1, gamma=0, and lambda=1 counts the number of independent sets. We define two notions, nearly-independent phase-correlated spins and symmetry breaking. We prove that it is #BIS-hard to approximate the partition function of any two-spin system on bipartite graphs supporting these two notions. As a consequence, we show that #BIS on graphs of degree at most 6 is as hard to approximate as #BIS~without degree bound. The degree bound 6 is the best possible as Weitz presented an FPTAS to count independent sets on graphs of maximum degree 5. This result extends to the hard-core model and to other anti-ferromagnetic two-spin models. In particular, for all antiferromagnetic two-spin systems, namely those satisfying beta*gamma<1, we prove that when the infinite (Delta-1)-ary tree lies in the non-uniqueness region then it is #BIS-hard to approximate the partition function on bipartite graphs of maximum degree Delta, except for the case beta=gamma and lambda=1. The exceptional case is precisely the antiferromagnetic Ising model without an external field, and we show that it has an FPRAS on bipartite graphs. Our inapproximability results match the approximability results of Li et al., who presented an FPTAS for general graphs of maximum degree Delta when the parameters lie in the uniqueness region.

The Moran process models the spread of mutations in populations on graphs. We investigate the absorption time of the process, which is the time taken for a mutation introduced at a randomly chosen vertex to either spread to the whole population, or to become extinct. It is known that the expected absorption time for an advantageous mutation is polynomial on an n-vertex undirected graph, which allows the behaviour of the process on undirected graphs to be analysed using the Markov chain Monte Carlo method. We show that this does not extend to directed graphs by exhibiting an infinite family of directed graphs for which the expected absorption time is exponential in the number of vertices. However, for regular directed graphs, we give the expected absorption time is blog n lower bound and an explicit quadratic upper bound. We exhibit families of graphs matching these bounds and give improved bounds for other families of graphs, based on isoperimetric number. Our results are obtained via stochastic dominations which we demonstrate by establishing a coupling in a related continuous-time model. The coupling also implies several natural domination results regarding the fixation probability of the original (discrete-time) process, resolving a conjecture of Shakarian, Roos and Johnson.

Given a symmetric DxD matrix M over {0, 1, *}, a list M-partition of a graph G is a partition of the vertices of G into D parts which are associated with the rows of M. The part of each vertex is chosen from a given list in such a way that no edge of G is mapped to a 0 in M and no non-edge of G is mapped to a 1 in M. Many important graph-theoretic structures can be represented as list M-partitions including graph colourings, split graphs and homogeneous sets, which arise in the proofs of the weak and strong perfect graph conjectures. Thus, there has been quite a bit of work on determining for which matrices M computations involving list M-partitions are tractable. This paper focuses on the problem of counting list M-partitions, given a graph G and given lists for each vertex of G. We give an algorithm that solves this problem in polynomial time for every (fixed) matrix M for which the problem is tractable. The algorithm relies on data structures such as sparse-dense partitions and sub cube decompositions to reduce each problem instance to a sequence of problem instances in which the lists have a certain useful structure that restricts access to portions of M in which the interactions of 0s and 1s is controlled. We show how to solve the resulting restricted instances by converting them into particular counting constraint satisfaction problems (#CSPs) which we show how to solve using a constraint satisfaction technique known as "arc-consistency". For every matrix M for which our algorithm fails, we show that the problem of counting list M-partitions is #P-complete. Furthermore, we give an explicit characterisation of the dichotomy theorem - counting list M-partitions is tractable (in FP) if and only if the matrix M has a structure called a derectangularising sequence. Finally, we show that the meta-problem of determining whether a given matrix has a derectangularising sequence is NP-complete.

A homomorphism from a graph G to a graph H is a function from V(G) to V(H) that preserves edges. Many combinatorial structures that arise in mathematics and computer science can be represented naturally as graph homomorphisms and as weighted sums of graph homomorphisms. In this paper, we study the complexity of counting homomorphisms modulo 2. The complexity of modular counting was introduced by Papadimitriou and Zachos and it has been pioneered by Valiant who famously introduced a problem for which counting modulo 7 is easy but counting modulo 2 is intractable. Modular counting provides a rich setting in which to study the structure of homomorphism problems. In this case, the structure of the graph H has a big influence on the complexity of the problem. Thus, our approach is graph-theoretic. We give a complete solution for the class of cactus graphs, which are connected graphs in which every edge belongs to at most one cycle. Cactus graphs arise in many applications such as the modelling of wireless sensor networks and the comparison of genomes. We show that, for some cactus graphs H, counting homomorphisms to H modulo 2 can be done in polynomial time. For every other fixed cactus graph H, the problem is complete for the complexity class +P which is a wide complexity class to which every problem in the polynomial hierarchy can be reduced (using randomised reductions). Determining which H lead to tractable problems can be done in polynomial time. Our result builds upon the work of Faben and Jerrum, who gave a dichotomy for the case in which H is a tree.

We study the complexity of computing the sign of the Tutte polynomial of a graph. As there are only three possible outcomes (positive, negative, and zero), this seems at first sight more like a decision problem than a counting problem. Surprisingly, however, there are large regions of the parameter space for which computing the sign of the Tutte polynomial is actually \#P-hard. As a trivial consequence, approximating the polynomial is also \#P-hard in this case. Thus, approximately evaluating the Tutte polynomial in these regions is as hard as exactly counting the satisfying assignments to a CNF Boolean formula. For most other points in the parameter space, we show that computing the sign of the polynomial is in FP, whereas approximating the polynomial can be done in polynomial time with an NP oracle. As a special case, we completely resolve the complexity of computing the sign of the chromatic polynomial---this is easily computable at q=2 and when $q\leq 32/27$, and is NP-hard to compute for all other values of the parameter q.

We study the complexity of approximation for a weighted counting constraint satisfaction problem #CSP(F). In the conservative case, where F contains all unary functions, a classification is known for the Boolean domain. We give a classification for problems with general finite domain. We define weak log-modularity and weak log-supermodularity, and show that #CSP(F) is in FP if F is weakly log-modular. Otherwise, it is at least as hard to approximate as #BIS, counting independent sets in bipartite graphs, which is believed to be intractable. We further sub-divide the #BIS-hard case. If F is weakly log-supermodular, we show that #CSP(F) is as easy as Boolean log-supermodular weighted #CSP. Otherwise, it is NP-hard to approximate. Finally, we give a trichotomy for the arity-2 case. Then, #CSP(F) is in FP, is #BIS-equivalent, or is equivalent to #SAT, the problem of approximately counting satisfying assignments of a CNF Boolean formula.

An important tool in the study of the complexity of Constraint Satisfaction Problems (CSPs) is the notion of a relational clone, which is the set of all relations expressible using primitive positive formulas over a particular set of base relations. Post's lattice gives a complete classification of all Boolean relational clones, and this has been used to classify the computational difficulty of CSPs. Motivated by a desire to understand the computational complexity of (weighted) counting CSPs, we develop an analogous notion of functional clones and study the landscape of these clones. One of these clones is the collection of log-supermodular (lsm) functions, which turns out to play a significant role in classifying counting CSPs. In the conservative case (where all nonnegative unary functions are available), we show that there are no functional clones lying strictly between the clone of lsm functions and the total clone (containing all functions). Thus, any counting CSP that contains a single nontrivial non-lsm function is computationally as hard to approximate as any problem in #P. Furthermore, we show that any nontrivial functional clone (in a sense that will be made precise) contains the binary function “implies”. As a consequence, in the conservative case, all nontrivial counting CSPs are as hard to approximate as #BIS, the problem of counting independent sets in a bipartite graph. Given the complexity-theoretic results, it is natural to ask whether the “implies” clone is equivalent to the clone of lsm functions. We use the Möbius transform and the Fourier transform to show that these clones coincide precisely up to arity 3. It is an intriguing open question whether the lsm clone is finitely generated. Finally, we investigate functional clones in which only restricted classes of unary functions are available.

We investigate the computational difficulty of approximating the partition function of the ferromagnetic Ising model on a regular matroid. Jerrum and Sinclair have shown that there is a fully polynomial randomized approximation scheme (FPRAS) for the class of graphic matroids. On the other hand, the authors have previously shown, subject to a complexity-theoretic assumption, that there is no FPRAS for the class of binary matroids, which is a proper superset of the class of graphic matroids. In order to map out the region where approximation is feasible, we focus on the class of regular matroids, an important class of matroids which properly includes the class of graphic matroids and is properly included in the class of binary matroids. Using Seymour's decomposition theorem, we give an FPRAS for the class of regular matroids.

We consider the Moran process, as generalized by Lieberman, Hauert and Nowak (Nature, 433:312–316, 2005). A population resides on the vertices of a finite, connected, undirected graph and, at each time step, an individual is chosen at random with probability proportional to its assigned “fitness” value. It reproduces, placing a copy of itself on a neighbouring vertex chosen uniformly at random, replacing the individual that was there. The initial population consists of a single mutant of fitness r > 0 placed uniformly at random, with every other vertex occupied by an individual of fitness 1. The main quantities of interest are the probabilities that the descendants of the initial mutant come to occupy the whole graph (fixation) and that they die out (extinction); almost surely, these are the only possibilities. In general, exact computation of these quantities by standard Markov chain techniques requires solving a system of linear equations of size exponential in the order of the graph so is not feasible. We show that, with high probability, the number of steps needed to reach fixation or extinction is bounded by a polynomial in the number of vertices in the graph. This bound allows us to construct fully polynomial randomized approximation schemes (FPRAS) for the probability of fixation (when r ≥ 1) and of extinction (for all r > 0).

Motivated by a desire to understand the computational complexity of counting constraint satisfaction problems (counting CSPs), particularly the complexity of approximation, we study functional clones of functions on the Boolean domain, which are analogous to the familiar relational clones constituting Post's lattice. One of these clones is the collection of log-supermodular (lsm) functions, which turns out to play a significant role in classifying counting CSPs. In our study, we assume that non-negative unary functions (weights) are available. Given this, we prove that there are no functional clones lying strictly between the clone of lsm functions and the total clone (containing all functions). Thus, any counting CSP that contains a single nontrivial non-lsm function is computationally as hard as any problem in #P. Furthermore, any non-trivial functional clone (in a sense that will be made precise below) contains the binary function "implies". As a consequence, all non-trivial counting CSPs (with non-negative unary weights assumed to be available) are computationally at least as difficult as #BIS, the problem of counting independent sets in a bipartite graph. There is empirical evidence that #BIS is hard to solve, even approximately.

We provide evidence that it is computationally difficult to approximate the partition function of the ferromagnetic q -state Potts model when q > 2. Specifically, we show that the partition function is hard for the complexity class #RHPi under approximation-preserving reducibility. Thus, it is as hard to approximate the partition function as it is to find approximate solutions to a wide range of counting problems, including that of determining the number of independent sets in a bipartite graph. Our proof exploits the first-order phase transition of the “random cluster” model, which is a probability distribution on graphs that is closely related to the q -state Potts model.

In the standard consensus problem there are n processes with possibly different input values and the goal is to eventually reach a point at which all processes commit to exactly one of these values. We are studying a slight variant of the consensus problem called the stabilizing consensus problem [2]. In this problem, we do not require that each process commits to a final value at some point, but that eventually they arrive at a common, stable value without necessarily being aware of that. This should work irrespective of the states in which the processes are starting. Our main result is a simple randomized algorithm called median rule that, with high probability, just needs O(log m log log n + log n) time and work per process to arrive at an almost stable consensus for any set of m legal values as long as an adversary can corrupt the states of at most √n processes at any time. Without adversarial involvement, just O(log n) time and work is needed for a stable consensus, with high probability. As a by-product, we obtain a simple distributed algorithm for approximating the median of n numbers in time O(log m log log n + log n) under adversarial presence.

This paper studies - from the perspective of efficient computation - a type of competition that is widespread throughout the plant and animal kingdoms, higher education, politics and artificial contests. In this setting, an agent gains utility from his relative performance (on some measurable criterion) against other agents, as opposed to his absolute performance. We model this situation using ranking games in which each strategy corresponds to a level of competitiveness, and incurs an upfront cost that is higher for more competitive strategies. We study the Nash equilibria of these games, and polynomial-time algorithms for computing them. For games in which there is no tie between agents' levels of competitiveness we give a polynomial-time algorithm for computing an exact equilibrium in the 2-player case, and a characterization of Nash equilibria that shows an interesting parallel between these games and unrestricted 2-player games in normal form. When ties are allowed, via a reduction from these games to a subclass of anonymous games, we give polynomial-time approximation schemes for two special cases: constant-sized set of strategies, and constant number of players. The latter result is improved to a fully polynomial-time approximation scheme when the constant number of players only compete to win the game, i.e. to be ranked first.

The degree of a CSP instance is the maximum number of times that a variable may appear in the scope of constraints. We consider the approximate counting problem for Boolean CSPs with bounded-degree instances, for constraint languages containing the two unary constant relations $\{0\}$ and $\{1\}$. When the maximum degree is at least $25$ we obtain a complete classification of the complexity of this problem. It is exactly solvable in polynomial-time if every relation in the constraint language is affine. It is equivalent to the problem of approximately counting independent sets in bipartite graphs if every relation can be expressed as conjunctions of $\{0\}$, $\{1\}$ and binary implication. Otherwise, there is no FPRAS unless $\NPtime = \RPtime$. For lower degree bounds, additional cases arise in which the complexity is related to the complexity of approximately counting independent sets in hypergraphs.

Partition functions, also known as homomorphism functions, form a rich family of graph invariants that contain combinatorial invariants such as the number of k-colorings or the number of independent sets of a graph and also the partition functions of certain “spin glass” models of statistical physics such as the Ising model. Building on earlier work by Dyer and Greenhill [Random Structures Algorithms, 17 (2000), pp. 260–289] and Bulatov and Grohe [Theoret. Comput. Sci., 348 (2005), pp. 148–186], we completely classify the computational complexity of partition functions. Our main result is a dichotomy theorem stating that every partition function is either computable in polynomial time or #P-complete. Partition functions are described by symmetric matrices with real entries, and we prove that it is decidable in polynomial time in terms of the matrix whether a given partition function is in polynomial time or #P-complete. While in general it is very complicated to give an explicit algebraic or combinatorial description of the tractable cases, for partition functions described by Hadamard matrices (these turn out to be central in our proofs) we obtain a simple algebraic tractability criterion, which says that the tractable cases are those “representable” by a quadratic polynomial over the field $\mathbb{F}_2$.

\emph{Partition functions}, also known as \emph{homomorphism functions}, form a rich family of graph invariants that contain combinatorial invariants such as the number of $k$-colourings or the number of independent sets of a graph and also the partition functions of certain ``spin glass'' models of statistical physics such as the Ising model. Building on earlier work by Dyer and Greenhill (2000) and Bulatov and Grohe (2005), we completely classify the computational complexity of partition functions. Our main result is a dichotomy theorem stating that every partition function is either computable in polynomial time or \#P-complete. Partition functions are described by symmetric matrices with real entries, and we prove that it is decidable in polynomial time in terms of the matrix whether a given partition function is in polynomial time or \#P-complete. While in general it is very complicated to give an explicit algebraic or combinatorial description of the tractable cases, for partition functions described by a Hadamard matrices --- these turn out to be central in our proofs --- we obtain a simple algebraic tractability criterion, which says that the tractable cases are those ``representable'' by a quadratic polynomial over the field $\ensuremath{\mathbb{F}_2}$.

This paper gives a dichotomy theorem for the complexity of computing the partition function of an instance of a weighted Boolean constraint satisfaction problem. The problem is parameterized by a finite set $\mathcal{F}$ of nonnegative functions that may be used to assign weights to the configurations (feasible solutions) of a problem instance. Classical constraint satisfaction problems correspond to the special case of 0,1-valued functions. We show that computing the partition function, i.e., the sum of the weights of all configurations, is $\text{{\sf FP}}^{\text{{\sf#P}}}$-complete unless either (1) every function in $\mathcal{F}$ is of “product type,” or (2) every function in $\mathcal{F}$ is “pure affine.” In the remaining cases, computing the partition function is in P.

This paper addresses the problem of fair equilibrium selection in graphical games. Our approach is based on the data structure called the best response policy, which was proposed by Kearns et al. [13] as a way to represent all Nash equilibria of a graphical game. In [9], it was shown that the best response policy has polynomial size as long as the underlying graph is a path. In this paper, we show that if the underlying graph is abounded-degree tree and the best response policy has polynomial size then there is an efficient algorithm which constructs a Nash equilibrium that guarantees certain payoffs to all participants. Another attractive solution concept is a Nash equilibrium that maximizes the social welfare. We show that, while exactly computing the latter is infeasible (we prove that solving this problem may involve algebraic numbers of an arbitrarily high degree), there exists an FPTAS for finding such an equilibrium as long as the best response policy has polynomial size. These two algorithms can be combined to produce Nash equilibria that satisfy various fairness criteria.

In set-system auctions, there are several overlapping teams of agents, and a task that can be completed by any of these teams. The auctioneer's goal is to hire a team and pay as little as possible. Examples of this setting include shortest-path auctions and vertex-cover auctions. Recently, Karlin, Kempe and Tamir introduced a new definition of frugality ratio for this problem. Informally, the "frugality ratio" is the ratio of the total payment of a mechanism to a desired payment bound. The ratio captures the extent to which the mechanism overpays, relative to perceived fair cost in a truthful auction. In this paper, we propose a new truthful polynomial-time auction for the vertex cover problem and bound its frugality ratio. We show that the solution quality is with a constant factor of optimal and the frugality ratio is within a constant factor of the best possible worst-case bound; this is the first auction for this problem to have these properties. Moreover, we show how to transform any truthful auction into a frugal one while preserving the approximation ratio. Also, we consider two natural modifications of the definition of Karlin et al., and we analyse the properties of the resulting payment bounds, such as monotonicity, computational hardness, and robustness with respect to the draw-resolution rule. We study the relationships between the different payment bounds, both for general set systems and for specific set-system auctions, such as path auctions and vertex-cover auctions. We use these new definitions in the proof of our main result for vertex-cover auctions via a bootstrapping technique, which may be of independent interest.

The Tutte polynomial of a graph G is a two-variable polynomial T(G;x,y) that encodes many interesting properties of the graph. We study the complexityof the following problem, for rationals x and y: take as input a graph G, and output a value which is a good approximation to T(G;x,y). We are interested in determining for which points (x,y) there is a fullypolynomial randomised approximation scheme (FPRAS) for T(G;x,y). Our main contribution is a substantial widening of the region known to benon-FPRASable.

It is known that if P and NP are different then there is an infinite hierarchy of different complexity classes that lie strictly between them. Thus, if P ≠ NP, it is not possible to classify NP using any finite collection of complexity classes. This situation has led to attempts to identify smaller classes of problems within NP where dichotomy results may hold: every problem is either in P or is NP-complete. A similar situation exists for counting problems. If P ≠#P, there is an infinite hierarchy in between and it is important to identify subclasses of #P where dichotomy results hold. Graph homomorphism problems are a fertile setting in which to explore dichotomy theorems. Indeed, Feder and Vardi have shown that a dichotomy theorem for the problem of deciding whether there is a homomorphism to a fixed directed acyclic graph would resolve their long-standing dichotomy conjecture for all constraint satisfaction problems. In this article, we give a dichotomy theorem for the problem of counting homomorphisms to directed acyclic graphs. Let H be a fixed directed acyclic graph. The problem is, given an input digraph G , determine how many homomorphisms there are from G to H . We give a graph-theoretic classification, showing that for some digraphs H , the problem is in P and for the rest of the digraphs H the problem is #P-complete. An interesting feature of the dichotomy, which is absent from previously known dichotomy results, is that there is a rich supply of tractable graphs H with complex structure.

Suppose that a set of m tasks are to be shared as equally as possible among a set of n resources. A game-theoretic mechanism to find a suitable allocation is to associate each task with a “selfish agent” and require each agent to select a resource, with the cost of a resource being the number of agents that select it. Agents would then be expected to migrate from overloaded to underloaded resources, until the allocation becomes balanced. Recent work has studied the question of how this can take place within a distributed setting in which agents migrate selfishly without any centralized control. In this paper we discuss a natural protocol for the agents which combines the following desirable features: It can be implemented in a strongly distributed setting, uses no central control, and has good convergence properties. For $m \gg n$, the system becomes approximately balanced (an $\epsilon$-Nash equilibrium) in expected time $O(\log \log m)$. We show using a martingale technique that the process converges to a perfectly balanced allocation in expected time $O(\log \log m + n^4)$. We also give a lower bound of $\Omega(\max\{\log \log m, n\})$ for the convergence time.

We consider Glauber dynamics on finite spin systems. The mixing time of Glauber dynamics can be bounded in terms of the influences of sites on each other. We consider three parameters bounding these influences: α, the total influence on a site, as studied by Dobrushin; α′, the total influence of a site, as studied by Dobrushin and Shlosman; and α″, the total influence of a site in any given context, which is related to the path-coupling method of Bubley and Dyer. It is known that if any of these parameters is less than 1 then random-update Glauber dynamics (in which a randomly chosen site is updated at each step) is rapidly mixing. It is also known that the Dobrushin condition α < 1 implies that systematic-scan Glauber dynamics (in which sites are updated in a deterministic order) is rapidly mixing. This paper studies two related issues, primarily in the context of systematic scan: (1) the relationship between the parameters α, α′ and α″, and (2) the relationship between proofs of rapid mixing using Dobrushin uniqueness (which typically use analysis techniques) and proofs of rapid mixing using path coupling. We use matrix balancing to show that the Dobrushin–Shlosman condition α′ < 1 implies rapid mixing of systematic scan. An interesting question is whether the rapid mixing results for scan can be extended to the α = 1 or α′ = 1 case. We give positive results for the rapid mixing of systematic scan for certain α = 1 cases. As an application, we show rapid mixing of systematic scan (for any scan order) for heat-bath Glauber dynamics for properq-colourings of a degree-Δ graphGwhenq≥ 2Δ.

Graphical games have been proposed as a game-theoretic model of large-scale distributed networks of non-cooperative agents. When the number of players is large, and the underlying graph has low degree, they provide a concise way to represent the players' payoffs. It has recently been shown that the problem of finding Nash equilibria in a general degree-3 graphical game with two actions per player is complete for the complexity class PPAD, indicating that it is unlikely that there is any polynomial-time algorithm for this problem. In this paper, we study the complexity of graphical games with two actions per player on bounded-degree trees. This setting was first considered by Kearns, Littman and Singh, who proposed a dynamic programming-based algorithm that computes all Nash equilibria of such games. The running time of their algorithm is exponential, though approximate equilibria can be computed efficiently. Later, Littman, Kearns and Singh proposed a modification to this algorithm that can find a single Nash equilibrium in polynomial time. We show that this modified algorithm is incorrect-the output is not always a Nash equilibrium. We then propose a new algorithm that is based on the ideas of Kearns et al. and computes all Nash equilibria in quadratic time if the input graph is a path, and in polynomial time if it is an arbitrary graph of maximum degree 2. Moreover, our algorithm can be used to compute Nash equilibria of graphical games on arbitrary trees, but the running time can be exponential, even when the tree has bounded degree. We show that this is inevitable -- any algorithm of this type will take exponential time, even on bounded-degree trees with pathwidth 2. It is an open question whether our algorithm runs in polynomial time on graphs with pathwidth 1, but we show that finding a Nash equilibrium for a 2-action graphical game in which the underlying graph has maximum degree 3 and constant pathwidth is PPAD-complete (so is unlikely to be tractable).

We consider the problem of sampling almost uniformly from the set of contingency tables with given row and column sums, when the number of rows is a constant. Cryan and Dyer [J. Comput. System Sci., 67 (2003), pp. 291-310] have recently given a fully polynomial randomized approximation scheme (fpras) for the related counting problem, which employs Markov chain methods indirectly. They leave open the question as to whether a natural Markov chain on such tables mixes rapidly. Here we show that the "2 x 2 heat-bath" Markov chain is rapidly mixing. We prove this by considering first a heat-bath chain operating on a larger window. Using techniques developed by Morris [Random Walks in Convex Sets, Ph.D. thesis, Department of Statistics, University of California, Berkeley, CA, 2000] and Morris and Sinclair [SIAM J. Comput., 34 (2004), pp. 195-226] for the multidimensional knapsack problem, we show that this chain mixes rapidly. We then apply the comparison method of Diaconis and Saloff-Coste [Ann. Appl. Probab., 3 (1993), pp. 696-730] to show that the 2 x 2 chain is also rapidly mixing.

Recursively-constructed couplings have been used in the past for mixing on trees. We show how to extend this technique to nontree-like graphs such as lattices. Using this method, we obtain the following general result. Suppose that G is a triangle-free graph and that for some $\degree \geq 3$, the maximum degree of G is at most $\degree$. We show that the spin system consisting of q-colorings of G has strong spatial mixing, provided $q > \alpha \degree-\gamma$, where $\alpha\approx 1.76322$ is the solution to $\alpha^\alpha=e$, and $\gamma = \frac{4\alpha^3-6\alpha^2-3\alpha+4}{2(\alpha^2-1)}\approx 0.47031$. Note that we have no additional lower bound on q or $\degree$. This is important for us because our main objective is to have results which are applicable to the lattices studied in statistical physics, such as the integer lattice $\zset^d$ and the triangular lattice. For these graphs (in fact, for any graph in which the distance-k neighborhood of a vertex grows subexponentially in k), strong spatial mixing implies that there is a unique infinite-volume Gibbs measure. That is, there is one macroscopic equilibrium rather than many. Our general result gives, for example, a ``hand proof' of strong spatial mixing for 7-colorings of triangle-free 4-regular graphs. (Computer-assisted proofs of this result were provided by Salas and Sokal [\textit{J. Stat. Phys}., 86 (1997), pp.\ 551--579] (for the rectangular lattice) and by Bubley, Dyer, Greenhill, and Jerrum [\textit{SIAM J. Comput.}, 29 (1999), pp.\ 387--400].) It also gives a hand proof of strong spatial mixing for 5-colorings of triangle-free 3-regular graphs. (A computer-assisted proof for the special case of the hexagonal lattice was provided earlier by Salas and Sokal [\textit{J. Stat. Phys}., 86 (1997), pp.\ 551--579].) Toward the end of the paper we show how to improve our general technique by considering the geometry of the lattice. The idea is to construct the recursive coupling from a system of recurrences rather than from a single recurrence. We use the geometry of the lattice to derive the system of recurrences. This gives us an analysis with a horizon of more than one level of induction, which leads to improved results. We illustrate this idea by proving strong spatial mixing for $q=10$ on the lattice $\zset^3$. Finally, we apply the idea to the triangular lattice, adding computational assistance. This gives us a (machine-assisted) proof of strong spatial mixing for $10$-colorings of the triangular lattice. (Such a proof for $11$ colors was given by Salas and Sokal [\textit{J. Stat. Phys}., 86 (1997), pp.\ 551--579].) For completeness, we also show that our strong spatial mixing proof implies rapid mixing of Glauber dynamics for sampling proper colorings of neighborhood-amenable graphs. (It is known that strong spatial mixing often implies rapid mixing, but existing proofs seem to be written for $\zset^d$.) Thus our strong spatial mixing results give rapid

Recursively-constructed couplings have been used in the past for mixing on trees. We show for the first time how to extend this technique to nontree-like graphs such as the integer lattice. Using this method, we obtain the following general result. Suppose that G is a triangle-free graph and that for some /spl Delta/ /spl ges/ 3, the maximum degree of G is at most /spl Delta/. We show that the spin system consisting of q-colourings of G has strong spatial mixing, provided q > /spl alpha//spl Delta/, where /spl alpha/ /spl ap/ 1.76322 is the solution to /spl alpha//sup /spl alpha// = e. Note that we have no additional lower bound on q or /spl Delta/. This is important for us because our main objective is to have results which are applicable to the lattices studied in statistical physics such as the integer lattice /spl Zopf//sup d/ and the triangular lattice. For these graphs (in fact, for any graph in which the distance-k neighbourhood of a vertex grows sub-exponentially in k), strong spatial mixing implies that there is a unique infinite-volume Gibbs measure. That is, there is one macroscopic equilibrium rather than many. We extend our general result, obtaining, for example, the first "hand proof" of strong spatial mixing for 7-colourings of triangle-free 4-regular graphs. (Computer-assisted proofs of this result were provided by Salas and Sokal (1997) for the rectangular lattice and by Bubley et al. (1999)). The extension also gives the first hand proof of strong spatial mixing for 5-colourings of triangle-free 3-regular graphs. (A computer-assisted proof for the special case of the hexagonal lattice was provided by Salas and Sokal). Towards the end of the paper we show how to improve our general technique by considering the geometry of the lattice. The idea is to construct the recursive coupling from a system of recurrences rather than from a single recurrence. We use the geometry of the lattice to derive the system of recurrences. This gives us an analysis with a horizon of more than one level of induction, which leads to improved results. We illustrate this idea by proving strong spatial mixing for q = 10 on the lattice /spl Zopf//sup 3/. Finally, we apply the idea to the triangular lattice, adding computational assistance. This gives us the first (machine-assisted) proof of strong spatial mixing for 10-colourings of the triangular lattice. (Such a proof for 11 colours was given by Salas and Sokal.).

We study contention-resolution protocols for multiple-access channels. We show that every backoff protocol is transient if the arrival rate, $\lambda$, is at least 0.42 and that the capacity of every backoff protocol is at most 0.42. Thus, we show that backoff protocols have (provably) smaller capacity than full-sensing protocols. Finally, we show that the corresponding results, with the larger arrival bound of 0.531, also hold for every acknowledgment-based protocol.

Cooper, Dyer, and Frieze [J. Algorithms, 39 (2001), pp. 117--134] studied the problem of sampling H-colorings (nearly) uniformly at random. Special cases of this problem include sampling colorings and independent sets and sampling from statistical physics models such as the Widom--Rowlinson model, the Beach model, the Potts model and the hard-core lattice gas model. Cooper et al. considered the family of "cautious" ergodic Markov chains with uniform stationary distribution and showed that, for every fixed connected "nontrivial " graph H, every such chain mixes slowly. In this paper, we give a complexity result for the problem. Namely, we show that for any fixed graph H with no trivial components, there is unlikely to be any polynomial almost uniform sampler (PAUS) for H-colorings. We show that if there were a PAUS for the H-coloring problem, there would also be a PAUS for sampling independent sets in bipartite graphs, and, by the self-reducibility of the latter problem, there would be a fully polynomial randomized approximation scheme (FPRAS) for #BIS---the problem of counting independent sets in bipartite graphs. Dyer, Goldberg, Greenhill, and Jerrum have shown that #BIS is complete in a certain logically defined complexity class. Thus, a PAUS for sampling H-colorings would give an FPRAS for the entire complexity class. In order to achieve our result we introduce the new notion of sampling-preserving reduction which seems to be more useful in certain settings than approximation-preserving reduction.

In this paper we consider the following scenario. A set of n jobs with different threads is being run concurrently. Each job has an associated weight, which gives the proportion of processor time that it should be allocated. In a single time quantum, p threads of (not necessarily distinct) jobs receive one unit of service, and we require a rule that selects those p threads, at each quantum. Proportionate fairness means that over time, each job will have received an amount of service that is proportional to its weight. That aim cannot be achieved exactly due to the discretisation of service provision, but we can still hope to bound the extent to which service allocation deviates from its target. It is important that any scheduling rule be simple since the rule will be used frequently.We consider a variant of the Surplus Fair Scheduling (SFS) algorithm of Chandra, Adler, Goyal, and Shenoy. Our variant, which is appropriate for scenarios where jobs consist of multiple threads, retains the properties that make SFS empirically attractive but allows the first proof of proportionate fairness in a multiprocessor context. We show that when the variant is run, no job lags more than p H(n)-p+1 steps below its target number of services, where H(n) is the Harmonic function. Also, no job is over-supplied by more than O(1) extra services. This analysis is tight and it also extends to an adversarial setting, which models some situations in which the relative weights of jobs change over time.

In this paper we analyze a very simple dynamic work-stealing algorithm. In the work-generation model, there are n (work) generators. A generator-allocation function is simply a function from the n generators to the n processors. We consider a fixed, but arbitrary, distribution $\cal D$ over generator-allocation functions. During each time step of our process, a generator-allocation function h is chosen from $\cal D$, and the generators are allocated to the processors according to h. Each generator may then generate a unit-time task, which it inserts into the queue of its host processor. It generates such a task independently with probability $\lambda$. After the new tasks are generated, each processor removes one task from its queue and services it. For many choices of $\cal D$, the work-generation model allows the load to become arbitrarily imbalanced, even when $\lambda < 1$. For example, $\cal D$ could be the point distribution containing a single function h which allocates all of the generators to just one processor. For this choice of $\cal D$, the chosen processor receives around $\lambda n$ units of work at each step and services one. The natural work-stealing algorithm that we analyze is widely used in practical applications and works as follows. During each time step, each empty processor (with no work to do) sends a request to a randomly selected other processor. Any nonempty processor having received at least one such request in turn decides (again randomly) in favor of one of the requests. The number of tasks which are transferred from the nonempty processor to the empty one is determined by the so-called work-stealing functionf . In particular, if a processor that accepts a request has $\ell$ tasks stored in its queue, then $f(\ell)$ tasks are transferred to the currently empty one. A popular work-stealing function is $f(\ell)=\lfloor \ell/2\rfloor$, which transfers (roughly) half of the tasks. We analyze the long-term behavior of the system as a function of $\lambda$ and f. We show that the system is stable for any constant generation rate $\lambda < 1$ and for a wide class of functions f. Most intuitively sensible functions are included in this class (for example, every monotonically nondecreasing function f which satisfies $0 \leq f(\ell)\leq \ell/2$ and $f(\ell)=\omega(1)$ as a function of $\ell$ is included). Furthermore, we give upper bounds on theaverage system load (as a function of f and n). Our proof techniques combine Lyapunov function arguments with domination arguments, which are needed to cope with dependency.

We consider the problem of sampling almost uniformly from the set of contingency tables with given row and column sums, when the number of rows is a constant. (2002) have recently given a fully polynomial randomized approximation scheme (fpras) for the related counting problem, which only employs Markov chain methods indirectly. But they leave open the question as to whether a natural Markov chain on such tables mixes rapidly. Here we answer this question in the affirmative, and hence provide a very different proof of the main result of Cryan and Dyer. We show that the "2 /spl times/ 2 heat-bath" Markov chain is rapidly mixing. We prove this by considering first a heat-bath chain operating on a larger window. Using techniques developed by Morris and Sinclair (2002) (see also Morris (2002)) for the multidimensional knapsack problem, we show that this chain mixes rapidly. We then apply the comparison method of Diaconis and Saloff-Coste (1993) to show that the 2 /spl times/ 2 chain is rapidly mixing. As part of our analysis, we give the first proof that the 2 /spl times/ 2 chain mixes in time polynomial in the input size when both the number of rows and the number of columns is constant.

Cooper, Dyer and Frieze studied the problem of sampling H-colourings (nearly) uniformly at random. Special cases of this problem include sampling colourings and independent sets and sampling from statistical physics models such as the Widom-Rowlinson model, the Beach model, the Potts model and the hard-core lattice gas model. Cooper et al. considered the family of "cautious" ergodic Markov chains with uniform stationary distribution and showed that, for every fixed connected "nontrivial" graph H, every such chain mixes slowly. In this paper, we give a complexity result for the problem. Namely, we show that for any fixed graph H with no trivial components, there is unlikely to be any Polynomial Almost Uniform Sampler (PAUS) for H-colourings. We show that if there were a PAUS for the H-colouring problem, there would also be a PAUS for sampling independent sets in bipartite graphs and, by the self-reducibility of the latter problem, there would be a Fully-Polynomial Randomised Approximation Scheme (FPRAS) for BIS --- the problem of counting independent sets in bipartite graphs. Dyer, Goldberg, Greenhill and Jerrum have shown that BIS is complete in a certain logically-defined complexity class. Thus, a PAUS for sampling H-colourings would give an FPRAS for the entire complexity class. In order to achieve our result we introduce the new notion of sampling-preserving reduction which seems to be more useful in certain settings than approximation-preserving reduction.

In this paper we analyse a very simple dynamic work-stealing algorithm. In the work-generation model, there are n generators which are arbitrarily distributed among a set of n processors. During each time-step, with probability /spl lambda/, each generator generates a unit-time task which it inserts into the queue of its host processor. After the new tasks are generated, each processor removes one task from its queue and services it. Clearly, the work-generation model allows the load to grow more and more imbalanced, so, even when /spl lambda/<1, the system load can be unbounded. The natural work-stealing algorithm that we analyse works as follows. During each time step, each empty processor sends a request to a randomly selected other processor. Any non-empty processor having received at least one such request in turn decides (again randomly) in favour of one of the requests. The number of tasks which are transferred from the non-empty processor to the empty one is determined by the so-called work-stealing function f. We analyse the long-term behaviour of the system as a function of /spl lambda/ and f. We show that the system is stable for any constant generation rate /spl lambda/<1 and for a wide class of functions f. We give a quantitative description of the functions f which lead to stable systems. Furthermore, we give upper bounds on the average system load (as a function of f and n).

The j-state general Markov model of evolution (due to Steel) is a stochastic model concerned with the evolution of strings over an alphabet of size j. In particular, the two-state general Markov model of evolution generalizes the well-known Cavender--Farris--Neyman model of evolution by removing the symmetry restriction (which requires that the probability that a "0" turns into a "1" along an edge is the same as the probability that a "1" turns into a "0" along the edge). Farach and Kannan showed how to probably approximately correct (PAC)-learn Markov evolutionary trees in the Cavender--Farris--Neyman model provided that the target tree satisfies the additional restriction that all pairs of leaves have a sufficiently high probability of being the same. We show how to remove both restrictions and thereby obtain the first polynomial-time PAC-learning algorithm (in the sense of Kearns et al. [Proceedings of the 26th Annual ACM Symposium on the Theory of Computing, 1994, pp. 273--282]) for the general class of two-state Markov evolutionary trees.

Job-shop scheduling is a classical NP-hard problem. Shmoys, Stein, and Wein presented the first polynomial-time approximation algorithm for this problem that has a good (polylogarithmic) approximation guarantee. We improve the approximation guarantee of their work and present further improvements for some important NP-hard special cases of this problem (e.g., in the preemptive case where machines can suspend work on operations and later resume). We also present NC algorithms with improved approximation guarantees for some NP-hard special cases.

We study contention resolution in a multiple-access channel such as the Ethernet channel. In the model that we consider,nusers generate messages for the channel according to a probability distribution. Raghavan and Upfal have given a protocol in which the expecteddelay(time to get serviced) of every message is O(logn) when messages are generated according to a Bernoulli distribution with generation rate up to about 1/10. Our main results are the following protocols: (a) one in which the expected average message delay is O(1) when messages are generated according to a Bernoulli distribution with a generation rate smaller than 1/e, and (b) one in which the expected delay of any message is O(1) for an analogous model in which users are synchronized (i.e., they agree about the time), there are potentially an infinite number of users, and messages are generated according to a Poisson distribution with generation rate up to 1/e. (Each message constitutes a new user.) To achieve (a), we first show how to simulate (b) usingnsynchronized users, and then show how to build the synchronization into the protocol.

A new method for analyzing the mixing time of Markov chains is described. This method is an extension of path coupling and involves analyzing the coupling over multiple steps.The expected behavior of the coupling at a certain stopping time is used to bound the expected behavior of the coupling after a fixed number of steps. The new method is applied to analyze the mixing time of the Glauber dynamics for graph colorings. We show that the Glauber dynamics has O(n log(n)) mixing time for triangle-free $\Delta$-regular graphs if k colors are used, where $k\geq (2-\eta)\Delta$, for some small positive constant $\eta$. This is the first proof of an optimal upper bound for the mixing time of the Glauber dynamics for some values of k in the range $k\leq 2\Delta$.

We give a polynomial-time algorithm for the following problem: Given a degree sequence in which each degree is bounded from above by a constant, select, uniformly at random, an unlabelled connected multigraph with the given degree sequence. We also give a polynomial-time algorithm for the following related problem: Given a molecular formula, select, uniformly at random, a structural isomer having the given formula.

We present a work-optimal randomized algorithm for simulating a shared memory machine (PRAM) on an optical communication parallel computer (OCPC). The OCPC model is motivated by the potential of optical communication for parallel computation. The memory of an OCPC is divided into modules, one module per processor. Each memory module only services a request on a timestep if it receives exactly one memory request. Our algorithm simulates each step of an n lg lg n-processor EREW PRAM on an n-processor OCPC in O(lg lg n) expected delay. (The probability that the delay is longer than this is at most $n^{-\alpha}$ for any constant $\alpha$.) The best previous simulation, due to Valiant, required $\Theta(\log n)$ expected delay.

The j-State General Markov Model of evolution M. Steel (1994) is a stochastic model concerned with the evolution of strings over an alphabet of size j. In particular, the Two-State General Markov Model of evolution generalises the well-known Cavender-Farris-Neyman model of evolution by removing the symmetry restriction (which requires that the probability that a '0'' turns into a '1' along an edge is the same as the probability that a '1' turns into a '0' along the edge). M. Farach and S. Kannan (1996) showed how to PAC-learn Markov Evolutionary Trees in the Cavender-Farris-Neyman model provided that the target tree satisfies the additional restriction that all pairs of leaves have a sufficiently high probability of being the same. We show how to remove both restrictions and thereby obtain the first polynomial-time PAC-learning algorithm (in the sense of Kearns et al.) for the general class of Two-State Markov Evolutionary Trees.

Opticalcommunication is likely to significantly speed up parallel computation because the vast bandwidth of the optical medium can be divided to produce communication networks of very high degree. However, the problem of contention in high-degree networks makes the routing problem in these networks theoretically (and practically) difficult. In this paper we examine Valiant's h-relation routing problem, which is a fundamental problem in the theory of parallel computing. The h-relation routing problem arises both in the direct implementation of specific parallel algorithms on distributed-memory machines and in the general simulation of shared-memory models such as the PRAM on distributed-memory machines. In an h-relation routing problem each processor has up to h messages that it wishes to send to other processors and each processor is the destination of at most h messages. We present a lower bound for routing an h-relation (for any h > 1) on a complete optical network of size n. Our lower bound applies to any randomized distributed algorithm for this task. Specifically, we show that the expected number of communication steps required to route an arbitrary h-relation is $\Omega(h + \sqrt{\,\log\log n}\,)$. This is the first known lower bound for this problem which does not restrict the class of algorithms under consideration.

We study contention resolution in multiple-access channels such as the Ethernet. Under a stochastic model of continuous packet generation from a set of n processors, we construct a protocol which guarantees constant expected delay for generation rates up to a fixed constant /spl lambda//sub 0/<1. Previous protocols which are stable for constant arrival rates do not guarantee constant expected delay. The two protocols that achieved results closest to this are one by Raghavan and Upfal, which only guarantees logarithmic (in n) expected delay, and one by Paterson and Srinivasan, which only guarantees constant expected delay with high probability. (In the latter protocol, there is a non-zero probability that the initial clock synchronization might fail and cause the expected delay to grow unboundedly.) Although those protocols do not guarantee constant expected delay, we have used ideas from them in the construction of our protocol, which does guarantee constant expected delay. We achieve our results using a technique called Robust Synchronization which is applied periodically in our protocol. The introduction of this technique and the analysis of this technique are the main contributions of the paper.

In this paper, we consider the problem of interprocessor communication on parallel computers that have optical communication networks. We consider the completely connected optical-communication parallel computer (OCPC), which has a completely connected optical network, and also the mesh-of-optical-buses parallel computer (MOB-PC), which has a mesh of optical buses as its communication network. The particular communication problem that we study is that of realizing an h-relation. In this problem, each processor has at most h messages to send and at most h messages to receive. It is clear that any 1-relation can be realized in one communication step on an OCPC. However, the best previously known p-processor OCPC algorithm for realizing an arbitrary h-relation for h > 1 requires $\Theta(h + \log p)$ expected communication steps. (This algorithm is due to Valiant and is based on earlier work of Anderson and Miller.) Valiant's algorithm is optimal only for $h=\Omega(\log p)$, and it is an open question of Geréb-Graus and Tsantilas whether there is a faster algorithm for h=o(log p). In this paper, we answer this question in the affirmative and we extend the range of optimality by considering the case in which $h\leq \log p$. In particular, we present a $\Theta(h + \log\log p)$-communication-step randomized algorithm that realizes an arbitrary h-relation on a p-processor OCPC. We show that if $h\leq \log p$, then the failure probability can be made as small as $p^{-\alpha}$ for any positive constant $\alpha$. We use the OCPC algorithm as a subroutine in a $\Theta(h + \log\log p)$-communication-step randomized algorithm that realizes an arbitrary h-relation on a $p\times p$-processor MOB-PC. Once again, we show that if $h\leq \log p$, then the failure probability can be made as small as $p^{-\alpha}$ for any positive constant $\alpha$.

Article Free Access Share on Analysis of a simple learning algorithm: learning foraging thresholds for lizards Author: Leslie Ann Goldberg Department of Computer Science, University of Warwick, Coventry CV4 7AL, United Kingdom Department of Computer Science, University of Warwick, Coventry CV4 7AL, United KingdomView Profile Authors Info & Claims COLT '96: Proceedings of the ninth annual conference on Computational learning theoryJanuary 1996 Pages 2–9https://doi.org/10.1145/238061.238063Published:01 January 1996Publication History 0citation178DownloadsMetricsTotal Citations0Total Downloads178Last 12 Months5Last 6 weeks1 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

Inferring phylogenetic trees is a fundamental problem in computational biology. We present a new objective criterion, the phylogenetic number, for evaluating evolutionary trees for species defined by biomolecular sequences or other qualitative characters. The phylogenetic number of a tree T is the maximum number of times that any given character state arises in T. By contrast, the classical parsimony criterion measures the total number of times that different character states arise in T. We consider the following related problems: finding the tree with minimum phylogenetic number, and computing the phylogenetic number of a given topology in which only the leaves are labeled by species. When the number of states is bounded (as is the case for biomolecular sequence characters), we can solve the second problem in polynomial time. Given the topology for an evolutionary tree, we can also compute a phylogeny with phylogenetic number 2 (when one exists) for an arbitrary number of states. This algorithm can be used to further distinguish trees that are equal under parsimony. We also consider a number of other related problems.

Optical communication is likely to significantly speed up parallel computation because the vast bandwidth of the optical medium can be divided to produce communication networks of very high degree. However, the problem of contention in high-degree networks makes the routing problem in these networks theoretically (and practically) difficult. In this paper we examine Valiant's h-relation routing problem, which is a fundamental problem in the theory of parallel computing. The h-relation routing problem arises both in the direct implementation of specific parallel algorithms on distributed-memory machines and in the general simulation of shared memory models such as the PRAM on distributed-memory machines. In an h

We present a work-optimal randomized algorithm for simulating a shared memory machine (PRAM) on an optical communication parallel computer (OCPC). The OCPC model is motivated by the potential of optical communication for parallel computation. The memory of an OCPC is divided into modules, one module per processor. Each memory module only services a request on a timestep if it receives exactly one memory request.

paper we consider the probcommunication on a Compltd ely Connected Optical Communication Parallel Computer (OCPC).The particular problem we study is that of realizing an h-r-elation.In this problem, each processor has at most h messages to send and at most h messages to receive.It is clear that any 1-relation can be realized in one communication step on an OCPC.However, the best known p-processor OCPC algorithm for realizing an arbitrary h-relation for h > 1 requires @(h + logp) expected communication steps.(This algorithm is due to Valiant and is based on earlier work of Anderson and Miller.) Valiant's algorithm is optimal only for h = f2(log p) and it is an open question of Ger6b-Graus and Tsantilas whether there is a faster algorithm for h = o(logp).In this paper we answer this question in the affirmative by presenting 1

Article Polynomial space polynomial delay algorithms for listing families of graphs Share on Author: Leslie Ann Goldberg View Profile Authors Info & Claims STOC '93: Proceedings of the twenty-fifth annual ACM symposium on Theory of ComputingJune 1993 Pages 218–225https://doi.org/10.1145/167088.167160Online:01 June 1993Publication History 9citation317DownloadsMetricsTotal Citations9Total Downloads317Last 12 Months5Last 6 weeks0 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteGet Access