For sink-free orientations in graphs of minimum degree at least $3$, we show that there is a deterministic approximate counting algorithm that runs in time $O((n^{73}/\varepsilon^{72})\log(n/\varepsilon))$, a near-linear time sampling algorithm, and a randomised approximate counting algorithm that runs in time $O((n/\varepsilon)^2\log(n/\varepsilon))$, where $n$ denotes the number of vertices of the input graph and $0<\varepsilon<1$ is the desired accuracy. All three algorithms are based on a local implementation of the sink popping method (Cohn, Pemantle, and Propp, 2002) under the partial rejection sampling framework (Guo, Jerrum, and Liu, 2019).

We show that the flip chain for non-crossing spanning trees of n+1 points in convex position mixes in time O(n⁸log n). We use connections between Fuss-Catalan structures to construct a comparison argument with a chain similar to Wilson’s lattice path chain (Wilson 2004).

We show that spin systems with bounded degrees and coupling independence admit fully polynomial time approximation schemes (FPTAS). We design a new recursive deterministic counting algorithm to achieve this. As applications, we give the first FPTASes for q-colourings on graphs of bounded maximum degree $\Delta \geq 3$, when $q \geq\left(11 / 6-\varepsilon_{0}\right) \Delta$ for some small $\varepsilon_{0} \approx 10^{-5}$, or when $\Delta \geq 125$ and $q \geq 1.809 \Delta$, and on graphs with sufficiently large (but constant) girth, when $q \geq \Delta+3$. These bounds match the current best randomised approximate counting algorithms by Chen, Delcourt, Moitra, Perarnau, and Postle (2019), Carlson and Vigoda (2024), and Chen, Liu, Mani, and Moitra (2023), respectively.

We give a deterministic polynomial-time approximation scheme (FPTAS) for the volume of the truncated fractional matching polytope for graphs of maximum degree $Δ$, where the truncation is by restricting each variable to the interval $[0,\frac{1+δ}Δ]$, and $δ\le \frac{C}Δ$ for some constant $C>0$. We also generalise our result to the fractional matching polytope for hypergraphs of maximum degree $Δ$ and maximum hyperedge size $k$, truncated by $[0,\frac{1+δ}Δ]$ as well, where $δ\le CΔ^{-\frac{2k-3}{k-1}}k^{-1}$ for some constant $C>0$. The latter result generalises both the first result for graphs (when $k=2$), and a result by Bencs and Regts (2024) for the truncated independence polytope (when $Δ=2$). Our approach is based on the cluster expansion technique.

Abstract. We present a new framework to derandomize certain Markov chain Monte Carlo (MCMC) algorithms. As in MCMC, we first reduce counting problems to sampling from a sequence of marginal distributions. For the latter task, we introduce a method called coupling toward the past that can, in logarithmic time, evaluate one or a constant number of variables from a stationary Markov chain state. Since there are at most logarithmic random choices, this leads to very simple derandomization. As an application, we provide an efficient deterministic approximate counting algorithm for hypergraph independent sets, under local lemma type conditions matching, up to lower-order factors, their state-of-the-art randomized counterparts.

We give a Õ(n) time almost uniform sampler for independent sets of a matroid, whose ground set has n elements and is given by an independence oracle. As a consequence, one can sample connected spanning subgraphs of a given graph G = (V,E) in Õ(|E|) time, whereas the previous best algorithm takes O(|E||V|) time. This improvement, in turn, leads to a faster running time on estimating all-terminal network reliability. Furthermore, we generalise this near-linear time sampler to the random cluster model with q ≤ 1.

We give a fully polynomial-time randomized approximation scheme (FPRAS) for two terminal reliability in directed acyclic graphs (DAGs). In contrast, we also show the complementing problem of approximating two terminal unreliability in DAGs is #BIS-hard.

We present two randomised approximate counting algorithms with Oe(n2−c/ε2) running time for some constant c > 0 and accuracy ε: 1. for the hard-core model with fugacity λ on graphs with maximum degree ∆ when λ = O(∆−1.5−c1) where c1 = c/(2 − 2c); 2. for spin systems with strong spatial mixing (SSM) on planar graphs with quadratic growth, such as Z2. For the hard-core model, Weitz’s algorithm (STOC, 2006) achieves sub-quadratic running time when correlation decays faster than the neighbourhood growth, namely when λ = o(∆−2). Our first algorithm does not require this property and extends the range where sub-quadratic algorithms exist. Our second algorithm appears to be the first to achieve sub-quadratic running time up to the SSM threshold, albeit on a restricted family of graphs. It also extends to (not necessarily planar) graphs with polynomial growth, such as Zd, but with a running time of the form O (n2ε−2/2c(log n)1/d) where d is the exponent of the polynomial growth and c > 0 is some constant.

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.

We present a new framework to derandomise certain Markov chain Monte Carlo (MCMC) algorithms. As in MCMC, we first reduce counting problems to sampling from a sequence of marginal distributions. For the latter task, we introduce a method called coupling towards the past that can, in logarithmic time, evaluate one or a constant number of variables from a stationary Markov chain state. Since there are at most logarithmic random choices, this leads to very simple derandomisation. We provide two applications of this framework, namely efficient deterministic approximate counting algorithms for hypergraph independent sets and hypergraph colourings, under local lemma type conditions matching, up to lower order factors, their state-of-the-art randomised counterparts.

We study the sampling problem for the ferromagnetic Ising model with consistent external fields, and in particular, Swendsen-Wang dynamics on this model. We introduce a new grand model unifying two closely related models: the subgraph world and the random cluster model. Through this new viewpoint, we show: polynomial mixing time bounds for Swendsen-Wang dynamics and (edge-flipping) Glauber dynamics of the random cluster model, generalising the bounds and simplifying the proofs for the no-field case by Guo and Jerrum (2018); near linear mixing time for the two dynamics above if the maximum degree is bounded and all fields are (consistent and) bounded away from 1.

We study the problem of sampling almost uniform proper q-colourings in k-uniform simple hypergraphs with maximum degree Δ. For any δ>0, if k≥20(1+δ)δ and q≥100Δ2+δk−4/δ−4, the running time of our algorithm is O~(poly(Δk)⋅n1.01), where n is the number of vertices. Our result requires fewer colours than previous results for general hypergraphs (Jain, Pham, and Voung, 2021; He, Sun, and Wu, 2021), and does not require Ω(logn) colours unlike the work of Frieze and Anastos (2017).

Intensive algorithmic efforts have been made to enable the rapid improvements of certificated robustness for complex ML models recently. However, current robustness certification methods are only able to certify under a limited perturbation radius. Given that existing pure data-driven statistical approaches have reached a bottleneck, in this paper, we propose to integrate statistical ML models with knowledge (expressed as logical rules) as a reasoning component using Markov logic networks (MLN), so as to further improve the overall certified robustness. This opens new research questions about certifying the robustness of such a paradigm, especially the reasoning component (e.g., MLN). As the first step towards understanding these questions, we first prove that the computational complexity of certifying the robustness of MLN is #P-hard. Guided by this hardness result, we then derive the first certified robustness bound for MLN by carefully analyzing different model regimes. Finally, we conduct extensive experiments on five datasets including both high-dimensional images and natural language texts, and we show that the certified robustness with knowledge-based logical reasoning indeed significantly outperforms that of the state-of-the-arts.

Abstract We prove a complexity classification for Holant problems defined by an arbitrary set of complex-valued symmetric constraint functions on Boolean variables. This is to specifically answer the question: Is the Fisher-Kasteleyn-Temperley (FKT) algorithm under a holographic transformation (Valiant, SIAM J. Comput. 37(5), 1565–1594 2008) a universal strategy to obtain polynomial-time algorithms for problems over planar graphs that are intractable on general graphs? There are problems that are #P-hard on general graphs but polynomial-time solvable on planar graphs. For spin systems (Kowalczyk 2010) and counting constraint satisfaction problems (#CSP) (Guo and Williams, J. Comput. Syst. Sci. 107, 1–27 2020), a recurring theme has emerged that a holographic reduction to FKT precisely captures these problems. Surprisingly, for Holant, we discover new planar tractable problems that are not expressible by a holographic reduction to FKT. In particular, a straightforward formulation of a dichotomy for planar Holant problems along the above recurring theme is false. A dichotomy theorem for #CSPd, which denotes #CSP where every variable appears a multiple of d times, has been an important tool in previous work. However the proof for the #CSPd dichotomy violates planarity, and it does not generalize to the planar case easily. In fact, due to our newly discovered tractable problems, the putative form of a planar #CSPd dichotomy is false when d ≥ 5. Nevertheless, we prove a dichotomy for planar #CSP2. In this case, the putative form of the dichotomy is true. (This is presented in Part II of the paper.) We manage to prove the planar Holant dichotomy relying only on this planar #CSP2 dichotomy, without resorting to a more general planar #CSPd dichotomy for d ≥ 3. A special case of the new polynomial-time computable problems is counting perfect matchings (#PM) over k-uniform hypergraphs when the incidence graph is planar and k ≥ 5. The same problem is #P-hard when k = 3 or k = 4, which is also a consequence of our dichotomy. When k = 2, it becomes #PM over planar graphs and is tractable again. More generally, over hypergraphs with specified hyperedge sizes and the same planarity assumption, #PM is polynomial-time computable if the greatest common divisor (gcd) of all hyperedge sizes is at least 5. It is worth noting that it is the gcd, and not a bound on hyperedge sizes, that is the criterion for tractability.

We extend the notion of spectral independence (introduced by Anari, Liu, and Oveis Gharan [ 4 ]) from the Boolean domain to general discrete domains. This property characterises distributions with limited correlations and implies that the corresponding Glauber dynamics is rapidly mixing. As a concrete application, we show that Glauber dynamics for sampling proper q -colourings mixes in polynomial-time for the family of triangle-free graphs with maximum degree Δ provided q ≥ ( α * + δ )Δ where α * ≈ 1.763 is the unique solution to α * = exp (1/ α * ) and δ Þ 0 is any constant. This is the first efficient algorithm for sampling proper q -colourings in this regime with possibly unbounded Δ. Our main tool of establishing spectral independence is the recursive coupling by Goldberg, Martin, and Paterson [ 25 ].

We extend the notion of spectral independence (introduced by Anari, Liu, and Oveis Gharan [2]) from the Boolean domain to general discrete domains. This property characterises distributions with limited correlations, and implies that the corresponding Glauber dynamics is rapidly mixing. As a concrete application, we show that Glauber dynamics for sampling proper q-colourings mixes in polynomial-time for the family of triangle-free graphs with maximum degree Δ provided q ≥ (α∗ + δ)Δ where α∗ ≈ 1.763 is the unique solution to α∗ = exp (1/α∗) and δ > 0 is any constant. This is the first efficient algorithm for sampling proper q-colourings in this regime with possibly unbounded Δ. Our main tool of establishing spectral independence is the recursive coupling by Goldberg, Martin, and Paterson [19].

We give new algorithms based on Markov chains to sample and approximately count satisfying assignments to k -uniform CNF formulas where each variable appears at most d times. For any k and d satisfying kd < n o(1) and k ≥ 20 log k + 20 log d + 60, the new sampling algorithm runs in close to linear time, and the counting algorithm runs in close to quadratic time. Our approach is inspired by Moitra (JACM, 2019), which remarkably utilizes the Lovász local lemma in approximate counting. Our main technical contribution is to use the local lemma to bypass the connectivity barrier in traditional Markov chain approaches, which makes the well-developed MCMC method applicable on disconnected state spaces such as SAT solutions. The benefit of our approach is to avoid the enumeration of local structures and obtain fixed polynomial running times, even if k = ω (1) or d = ω (1).

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 present fully polynomial-time (deterministic or randomised) approximation schemes for Holant problems, defined by a non-negative constraint function satisfying a generalised second-order recurrence modulo in a couple of exceptional cases. As a consequence, any non-negative Holant problem on cubic graphs has an efficient approximation algorithm unless the problem is equivalent to approximately counting perfect matchings, a central open problem in the area. This is in sharp contrast to the computational phase transition shown by two-state spin systems on cubic graphs. Our main technique is the recently established connection between zeros of graph polynomials and approximate counting.

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 zeros of the partition functions of ferromagnetic 2-state spin systems in terms of the external field, and obtain new zero-free regions of these systems via a refinement of Asano's and Ruelle's contraction method. The strength of our results is that they do not depend on the maximum degree of the underlying graph. Via Barvinok's method, we also obtain new efficient and deterministic approximate counting algorithms. When the edge interaction is attractive for both spins, our algorithm outperforms all other methods such as Markov chain Monte Carlo and correlation decay.

We give new algorithms based on Markov chains to sample and approximately count satisfying assignments to k-uniform CNF formulas where each variable appears at most d times. For any k and d satisfying kd

We show that the modified log-Sobolev constant for a natural Markov chain which converges to an $r$-homogeneous strongly log-concave distribution is at least $1/r$. Applications include a sharp mixing time bound for the bases-exchange walk for matroids, and a concentration bound for Lipschitz functions over these distributions.

We present fully polynomial-time (deterministic or randomised) approximation schemes for Holant problems, defined by a non-negative constraint function satisfying a generalised second order recurrence modulo a couple of exceptional cases. As a consequence, any non-negative Holant problem on cubic graphs has an efficient approximation algorithm unless the problem is equivalent to approximately counting perfect matchings, a central open problem in the area. This is in sharp contrast to the computational phase transition shown by 2-state spin systems on cubic graphs. Our main technique is the recently established connection between zeros of graph polynomials and approximate counting. We also use the “winding” technique to deduce the second result on cubic graphs.

We propose a new algorithmic framework, called partial rejection sampling , to draw samples exactly from a product distribution, conditioned on none of a number of bad events occurring. Our framework builds new connections between the variable framework of the Lovász Local Lemma and some classical sampling algorithms such as the cycle-popping algorithm for rooted spanning trees. Among other applications, we discover new algorithms to sample satisfying assignments of k -CNF formulas with bounded variable occurrences.

We give a fully polynomial-time approximation scheme (FPTAS) to count the number of $q$-colorings for $k$-uniform hypergraphs with maximum degree $\Delta$ if $k\ge 28$ and $q > 357 \Delta^{\frac{14}{k-14}}$. We also obtain a polynomial-time almost uniform sampler if $q>931 \Delta^{\frac{16}{k-16/3}}$. These are the first approximate counting and sampling algorithms in the regime $q\ll\Delta$ (for large $\Delta$ and $k$) without any additional assumptions. Our method is based on the recent work of Moitra (STOC, 2017). One important contribution of ours is to remove the dependency of $k$ and $\Delta$ in Moitra's approach.

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.

We give a fully polynomial-time randomized approximation scheme (FPRAS) for the all-terminal network reliability problem, which is to determine the probability that in an undirected graph, assuming each edge fails independently, the remainder of the graph is still connected. Our main contribution is to confirm a conjecture by Gorodezky and Pak [ Random Structures Algorithms, 44 (2014), pp. 201--223] that the expected running time of the “cluster-popping” algorithm in bidirected graphs is bounded by a polynomial in the size of the input.

For Markov chain Monte Carlo methods, one of the greatest discrepancies between theory and system is the scan order — while most theoretical development on the mixing time analysis deals with random updates, real-world systems are implemented with systematic scans. We bridge this gap for models that exhibit a bipartite structure, including, most notably, the Restricted/Deep Boltzmann Machine. The de facto implementation for these models scans variables in a layer-wise fashion. We show that the Gibbs sampler with a layerwise alternating scan order has its relaxation time (in terms of epochs) no larger than that of a random-update Gibbs sampler (in terms of variable updates). We also construct examples to show that this bound is asymptotically tight. Through standard inequalities, our result also implies a comparison on the mixing times.

We give a fully polynomial-time randomized approximation scheme (FPRAS) for the all-terminal network reliability problem, which is to determine the probability that, in a undirected graph, assuming each edge fails independently, the remaining graph is still connected. Our main contribution is to confirm a conjecture by Gorodezky and Pak (Random Struct. Algorithms, 2014), that the expected running time of the "cluster-popping" algorithm in bi-directed graphs is bounded by a polynomial in the size of the input.

We present a perfect simulation of the hard disks model via the partial rejection sampling method. Provided the density of disks is not too high, the method produces exact samples in O(log n) rounds, where n is the expected number of disks. The method extends easily to the hard spheres model in d>2 dimensions. In order to apply the partial rejection method to this continuous setting, we provide an alternative perspective of its correctness and run-time analysis that is valid for general state spaces.

We give a fully polynomial-time approximation scheme (FPTAS) to count the number of q-colorings for k-uniform hypergraphs with maximum degree Δ if k≥ 28 and q > 315Δ14/k−14. We also obtain a polynomial-time almost uniform sampler if q>798Δ16/k−16/3. These are the first approximate counting and sampling algorithms in the regime q≪Δ (for large Δ and k) without any additional assumptions. Our method is based on the recent work of Moitra (STOC, 2017). One important contribution of ours is to remove the dependency of k and Δ in Moitra’s approach.

We show for the first time that the mixing time of Glauber (single edge update) dynamics for the random cluster model at q = 2 is bounded by a polynomial in the size of the underlying graph. As a consequence, the Swendsen- Wang algorithm for the ferromagnetic Ising model at any temperature has the same polynomial mixing time bound.

We propose a new algorithmic framework, called “partial rejection sampling”, to draw samples exactly from a product distribution, conditioned on none of a number of bad events occurring. Our framework builds (perhaps surprising) new connections between the variable framework of the Lovász Local Lemma and some clas- sical sampling algorithms such as the “cycle-popping” algorithm for rooted spanning trees by Wilson. Among other applications, we discover new algorithms to sample satisfying assignments of k-CNF formulas with bounded variable occurrences.

For anti-ferromagnetic 2-spin systems, a beautiful connection has been established, namely that the following three notions align perfectly: the uniqueness of Gibbs measures in infinite regular trees, the decay of correlations (also known as spatial mixing), and the approximability of the partition function. The uniqueness condition implies spatial mixing, and an FPTAS for the partition function exists based on spatial mixing. On the other hand, non-uniqueness implies some long range correlation, based on which NP-hardness reductions are built. These connections for ferromagnetic 2-spin systems are much less clear, despite their similarities to anti-ferromagnetic systems. The celebrated Jerrum-Sinclair Markov chain [JS93] works even if spatial mixing fails. Also, for a fixed degree the uniqueness condition is non-monotone with respect to the external field, which seems to have no meaningful interpretation in terms of computational complexity. However, it is still intriguing whether there are some relationship underneath the apparent disparities among them. We provide some answers to this question. Let ; be the (0; 0) and (1; 1) edge interactions respectively ( > 1), and the external field for spin “0”. For graphs with degree bound Δ Δc + 1 where Δc = p p +1 -1 , regardless of the field (even inconsistent fields are allowed), correlation decay always holds and FPTAS exists. If all fields satisfy < c (assuming ), where c = ( =) Δc+1 2 , then a weaker version of spatial mixing holds in all trees. Moreover, if 1, then < c is sufficient to guarantee strong spatial mixing and FPTAS. This improves the best previous algorithm, a Markov chain based FPRAS for = [LLZ14]. The bound c is almost optimal and can be viewed as a variant of the uniqueness condition with the degree d relaxed to be a real number instead of an integer. When 1, uniqueness holds in all infinite regular trees, if and only if int c , where int c = ( =) ⌈Δc⌉+1 2 . If we allow fields > int c ′, where int c ′ = ( =) ⌊Δc⌋+2 2 , then approximating the partition function is #BIS-hard. Interestingly, unless Δc is an integer, neither c nor int c is the tight bound in each own respect. We provide examples where correlation decay continues to hold in a small interval beyond c, and irregular trees in which spatial mixing fails for some < int c .

We prove a complexity dichotomy theorem for Holant problems over an arbitrary set of complex-valued symmetric constraint functions $\mathcal{F}$ on Boolean variables. This extends and unifies all previous dichotomies for Holant problems on symmetric constraint functions (taking values without a finite modulus). We define and characterize all symmetric vanishing signatures; they turn out to be essential to the complete classification of Holant problems. The dichotomy theorem has an explicit tractability criterion expressible in terms of holographic transformations. A Holant problem defined by a set of constraint functions $\mathcal{F}$ is solvable in polynomial time if it satisfies this tractability criterion, and is #P-hard otherwise. The tractability criterion can be intuitively stated as follows: A set $\mathcal{F}$ is tractable if (1) every function in $\mathcal{F}$ has arity at most two; or (2) $\mathcal{F}$ is transformable to an affine type; or (3) $\mathcal{F}$ is transformable to a product type; or (4) $\mathcal{F}$ is vanishing, combined with the right type of binary functions; or (5) $\mathcal{F}$ belongs to a special category of vanishing-type Fibonacci gates. The proof of this theorem utilizes many previous dichotomy theorems on Holant problems and Boolean constraint satisfaction problems (#CSP). Holographic transformations play an indispensable role as both a proof technique and in the statement of the tractability criterion.

We prove a complexity dichotomy for complex-weighted Holant problems with an arbitrary set of symmetric constraint functions on Boolean variables. In the study of counting complexity, such as #CSP, there are problems which are #P-hard over general graphs but P-time solvable over planar graphs. A recurring theme has been that a holographic reduction [36] to FKT precisely captures these problems. This dichotomy answers the question: Is this a universal strategy? Surprisingly, we discover new planar tractable problems in the Holant framework (which generalizes #CSP) that are not expressible by a holographic reduction to FKT. In particular, the putative form of a dichotomy for planar Holant problems is false. Nevertheless, we prove a dichotomy for #CSP2, a variant of #CSP where every variable appears even times, that the presumed universality holds for #CSP2. This becomes an important tool in the proof of the full dichotomy, which refutes this universality in general. The full dichotomy says that the new P-time algorithms and the strategy of holographic reductions to FKT together are universal for these locally defined counting problems. As a special case of our new planar tractable problems, counting perfect matchings (#PM) over k-uniform hypergraphs is P-time computable when the incidence graph is planar and k ≥ 5. The same problem is #P-hard when k = 3 or k = 4, also a consequence of the dichotomy. More generally, over hypergraphs with specified hyperedge sizes and the same planarity assumption, #PM is P-time computable if the greatest common divisor (gcd) of all hyperedge sizes is at least 5.

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.

We show that an effective version of Siegel's Theorem on finiteness of integer solutions for a specific algebraic curve and an application of elementary Galois theory are key ingredients in a complexity classification of some Holant problems. These Holant problems, denoted by Holant(f), are defined by a symmetric ternary function f that is invariant under any permutation of the κ ≥ 3 domain elements. We prove that Holant(f) exhibits a complexity dichotomy. The hardness, and thus the dichotomy, holds even when restricted to planar graphs. A special case of this result is that counting edge κ-colorings is #P-hard over planar 3-regular multigraphs for all κ ≥ 3. In fact, we prove that counting edge κ-colorings is #P-hard over planar r-regular multigraphs for all κ ≥ r ≥ 3. The problem is polynomial-time computable in all other parameter settings. The proof of the dichotomy theorem for Holant(f) depends on the fact that a specific polynomial p(x, y) has an explicitly listed finite set of integer solutions, and the determination of the Galois groups of some specific polynomials. In the process, we also encounter the Tutte polynomial, medial graphs, Eulerian partitions, Puiseux series, and a certain lattice condition on the (logarithm of) the roots of polynomials.

We prove a complexity dichotomy theorem for Holant problems over an arbitrary set of complex-valued symmetric constraint functions {F} on Boolean variables. This extends and unifies all previous dichotomies for Holant problems on symmetric constraint functions (taking values without a finite modulus). We define and characterize all symmetric vanishing signatures. They turned out to be essential to the complete classification of Holant problems. The dichotomy theorem has an explicit tractability criterion. A Holant problem defined by a set of constraint functions {F} is solvable in polynomial time if it satisfies this tractability criterion, and is #P-hard otherwise. The tractability criterion can be intuitively stated as follows: A set {F} is tractable if (1) every function in {F} has arity at most two, or (2) {F} is transformable to an affine type, or (3) {F} is transformable to a product type, or (4) {F} is vanishing, combined with the right type of binary functions, or (5) {F} belongs to a special category of vanishing type Fibonacci gates. The proof of this theorem utilizes many previous dichotomy theorems on Holant problems and Boolean #CSP. Holographic transformations play an indispensable role, not only as a proof technique, but also in the statement of the dichotomy criterion.

For certain subclasses of NP, $\oplus$P, or #P characterized by local constraints, it is known that if there exist any problems within that subclass that are not polynomial time computable, then all the problems in the subclass are NP-complete, $\oplus$P-complete, or #P-complete. Such dichotomy results have been proved for characterizations such as constraint satisfaction problems and directed and undirected graph homomorphism problems, often with additional restrictions. Here we give a dichotomy result for the more expressive framework of Holant problems. For example, these additionally allow for the expression of matching problems, which have had pivotal roles in the development of complexity theory. As our main result we prove the dichotomy theorem that, for the class $\oplus$P, every set of symmetric Holant signatures of any arities that is not polynomial time computable is $\oplus$P-complete. The result exploits some special properties of the class $\oplus$P and characterizes four distinct tractable subclasses within $\oplus$P. It leaves open the corresponding questions for NP, $\#$P, and $\#_k$P for $k\neq 2$.

We prove a complexity dichotomy theorem for counting weighted Boolean CSP modulo k for any positive integer $k>1$. This generalizes a theorem by Faben for the unweighted setting. In the weighted setting, there are new interesting tractable problems. We first prove a dichotomy theorem for the finite field case where k is a prime. It turns out that the dichotomy theorem for the finite field is very similar to the one for the complex weighted Boolean #CSP, found by [Cai, Lu and Xia, STOC 2009]. Then we further extend the result to an arbitrary integer k.