The complexity of bilinear maps (equivalently, of 3-mode tensors) has been studied extensively, most notably in the context of matrix multiplication. While circuit complexity and tensor rank coincide asymptotically for 3-mode tensors, this correspondence breaks down for d ≥ 4 modes. As a result, the complexity of d-mode tensors for larger fixed d remains poorly understood, despite its relevance, e.g., in fine-grained complexity. Our paper explores this intermediate regime. First, we give a "graph-theoretic" proof of Strassen’s 2ω/3 bound on the asymptotic rank exponent of 3-mode tensors. Our proof directly generalizes to an upper bound of (d-1)ω/3 for d-mode tensors. Using refined techniques available only for d ≥ 4 modes, we improve this bound beyond the current state of the art for ω. We also obtain a bound of d/2+1 on the asymptotic exponent of circuit complexity of generic d-mode tensors and optimized bounds for d ∈ {4,5}. To the best of our knowledge, asymptotic circuit complexity (rather than rank) of tensors has not been studied before. To obtain a robust theory, we first ask whether low complexity of T and U imply low complexity of their Kronecker product T ⊗ U. While this crucially holds for rank (and thus for circuit complexity in 3 modes), we show that assumptions from fine-grained complexity rule out such a submultiplicativity for the circuit complexity of tensors with many modes. In particular, assuming the Hyperclique Conjecture, this failure occurs already for d = 8 modes. Nevertheless, we can salvage a restricted notion of submultiplicativity. From a technical perspective, our proofs heavily make use of the graph tensors T_H, as employed by Christandl and Zuiddam (Comput. Complexity 28 (2019) 27-56) and Christandl, Vrana and Zuiddam (Comput. Complexity 28 (2019) 57-111), whose modes correspond to the vertices of undirected graphs H. We make the simple but conceptually crucial observation that Kronecker products T_G ⊗ T_H are isomorphic to T_{G+H}, and that G and H may also be fractional graphs. By asymptotically converting generic tensors to specific graph tensors, we can use nontrivial results from algorithmic graph theory to study the rank and complexity of d-mode tensors for fixed d.

A problem dating back to Boole [Laws of Thought, Walton & Maberly,1854] is what can be computed about the probability of a finite union of events when given as input the probabilities of intersections of some of the events. The modern geometric study of the problem can be traced back to Hailperin [Amer. Math. Monthly 2 (1965) 343--359] who phrased the problem in the language of linear programming and generalized it to logical formulas of the events other than disjunction, heralding a substantial body of work in probabilistic logic [Nilsson, Artif.\ Intell.\ 28 (1986) 71--87], including the probabilistic satisfiability problem of Georgakopoulos, Kavvadis, and Papadimitriou [J.Complexity 4 (1988) 1--11], as well as fundamental connections to the geometry of metrics via cut and correlation polytopes [Deza and Laurent, Geometry of Cuts and Metrics, Springer, 1997] and to the study of marginal polytopes in graphical models of machine learning [Wainwright and Jordan, Found.\ Trends Mach.\ Learn. 1 (2008) 1--305]. This paper (i) describes the pertinent geometry of Boole's problem via coordinate projections of an elementary polytope arising essentially from Hailperin's linear program on the atoms of a Venn diagram, and (ii) shows that computing the optimal interval for the union probability is NP-hard, resolving an apparent gap in the literature highlighted by Pitowsky [Math.\ Programming 50 (1991) 395--414] and Boros et al. [Math.\ Oper.\ Res. 39 (2014) 1311--1329 and 51 (2026) 134--148].

We show that sufficiently low tensor rank for the balanced tripartitioning tensor P_d(x,y,z) = ∑_{A,B,C ∈ binom([3d],d):A∪ B∪ C = [3d]} x_A y_B z_C for a large enough constant d implies uniform arithmetic circuits for the matrix permanent that are exponentially smaller than circuits obtainable from Ryser’s formula. Under the same low-rank assumption, we obtain exponential-time improvements over the state of the art for a wide variety of related counting and decision problems. Our main methodological contribution is that the tensors P_n have a desirable Kronecker scaling property: They can be decomposed efficiently into a small sum of restrictions of Kronecker powers of P_d for constant d. We prove this with a new technique relying on Steinitz’s lemma, which we hence call Steinitz balancing. As a consequence of our methods, we show that the mentioned low-rank assumption (and hence the improved algorithms) is implied by Strassen’s asymptotic rank conjecture [Progr. Math. 120 (1994)], a bold conjecture that has recently seen intriguing progress.

The exponent σ(T) of a tensor T ∈ 𝔽^d⊗𝔽^d⊗𝔽^d over a field 𝔽 captures the base of the exponential growth rate of the tensor rank of T under Kronecker powers. Tensor exponents are fundamental from the standpoint of algorithms and computational complexity theory; for example, the exponent ω of square matrix multiplication can be characterized as ω = 2σ(MM₂), where MM₂ ∈ 𝔽⁴⊗𝔽⁴⊗𝔽⁴ is the tensor that represents 2×2 matrix multiplication. Strassen [FOCS 1986] initiated a duality theory for spaces of tensors that enables one to characterize the exponent of a tensor via objects in a dual space, called the asymptotic spectrum of the primal (tensor) space. While Strassen’s theory has considerable generality beyond the setting of tensors - Wigderson and Zuiddam [Asymptotic Spectra: Theory, Applications, and Extensions, preprint, 2023] give a recent exposition - progress in characterizing the dual space in the tensor setting has been slow, with the first universal points in the dual identified by Christandl, Vrana, and Zuiddam [J. Amer. Math. Soc. 36 (2023)]. In parallel to Strassen’s theory, the algebraic geometry community has developed a geometric theory of tensors aimed at characterizing the structure of the primal space and tensor exponents therein; the latter study was motivated in particular by an observation of Strassen (implicit in [J. Reine Angew. Math. 384 (1988)]) that matrix-multiplication tensors have limited universality in the sense that σ(𝔽^d⊗𝔽^d⊗𝔽^d) ≤ 2ω/3 = 4/3σ(MM₂) holds for all d ≥ 1. In particular, this limited universality of the tensor MM₂ puts forth the question whether one could construct explicit universal tensors that exactly characterize the worst-case tensor exponent in the primal space. Such explicit universal objects would, among others, give means towards a proof or a disproof of Strassen’s asymptotic rank conjecture [Progr. Math. 120 (1994)]; the former would immediately imply ω = 2 and, among others, refute the Set Cover Conjecture (cf. Björklund and Kaski [STOC 2024] and Pratt [STOC 2024]). Our main result is an explicit construction of a sequence 𝒰_d of zero-one-valued tensors that is universal for the worst-case tensor exponent; more precisely, we show that σ(𝒰_d) = σ(d) where σ(d) = sup_{T ∈ 𝔽^d⊗𝔽^d⊗𝔽^d}σ(T). We also supply an explicit universal sequence 𝒰_Δ localised to capture the worst-case exponent σ(Δ) of tensors with support contained in Δ ⊆ [d]×[d]×[d]; by combining such sequences, we obtain a universal sequence 𝒯_d such that σ(𝒯_d) = 1 holds if and only if Strassen’s asymptotic rank conjecture holds for d. Finally, we show that the limit lim_{d → ∞}σ(d) exists and can be captured as lim_{d → ∞} σ(D_d) for an explicit sequence (D_d)_{d = 1}^∞ of tensors obtained by diagonalisation of the sequences 𝒰_d. As our second result we relate the absence of polynomials of fixed degree vanishing on tensors of low rank, or more generally asymptotic rank, with upper bounds on the exponent σ(d). Using this technique, one may bound asymptotic rank for all tensors of a given format, knowing enough specific tensors of low asymptotic rank.

In this paper we further explore the recently discovered connection by Björklund and Kaski [STOC 2024] and Pratt [STOC 2024] between the asymptotic rank conjecture of Strassen [Progr. Math. 1994] and the three-way partitioning problem. We show that under the asymptotic rank conjecture, the chromatic number of an n-vertex graph can be computed deterministically in O (1.99982n ) time, thus giving a conditional answer to a question of Zamir [ICALP 2021], and questioning the optimality of the 2n poly(n ) time algorithm for chromatic number by Björklund, Husfeldt, and Koivisto [SICOMP 2009].

Strassen’s asymptotic rank conjecture [Progr. ‍Math. ‍120 ‍(1994)] claims a strong submultiplicative upper bound on the rank of a three-tensor obtained as an iterated Kronecker product of a constant-size base tensor. The conjecture, if true, most notably would put square matrix multiplication in quadratic time. We note here that some more-or-less unexpected algorithmic results in the area of exponential-time algorithms would also follow. Specifically, we study the so-called set cover conjecture, which states that for any є>0 there exists a positive integer constant k such that no algorithm solves the k-Set Cover problem in worst-case time ((2−є)n|F|poly(n)). The k-Set Cover problem asks, given as input an n-element universe U, a family F of size-at-most-k subsets of U, and a positive integer t, whether there is a subfamily of at most t sets in F whose union is U. The conjecture was formulated by Cygan, Fomin, Kowalik, Lokshtanov, Marx, Pilipczuk, Pilipczuk, and Saurabh ‍in the monograph Parameterized Algorithms [Springer, ‍2015], but was implicit as a hypothesis already in Cygan, Dell, Lokshtanov, Marx, Nederlof, Okamoto, Paturi, Saurabh, and Wahlstr'om ‍[CCC ‍2012, ACM ‍Trans. ‍Algorithms ‍2016], there conjectured to follow from the Strong Exponential Time Hypothesis. We prove that if the asymptotic rank conjecture is true, then the set cover conjecture is false. Using a reduction by Krauthgamer and Trabelsi [STACS ‍2019], in this scenario we would also get an ((2−δ)n)-time randomized algorithm for some constant δ>0 for another well-studied problem for which no such algorithm is known, namely that of deciding whether a given n-vertex directed graph has a Hamiltonian cycle. At a fine-grained level, our results do not need the full strength of the asymptotic rank conjecture; it suffices that the conclusion of the conjecture holds approximately for a single 7× 7× 7 tensor.

Monte Carlo integration is a key technique for designing randomized approximation schemes for counting problems, with applications, e.g., in machine learning and statistical physics. The technique typically enables massively parallel computation, however, with the risk that some of the delegated computations contain spontaneous or adversarial errors. We present an orchestration of the computations such that the outcome is accompanied with a proof of correctness that can be verified with substantially less computational resources than it takes to run the computations from scratch with state-of-the-art algorithms. Specifically, we adopt an algebraic proof system developed in computational complexity theory, in which the proof is represented by a polynomial; evaluating the polynomial at a random point amounts to a verification of the proof with probabilistic guarantees. We give examples of known Monte Carlo estimators that admit verifiable extensions with moderate computational overhead: for the permanent of zero--one matrices, for the model count of disjunctive normal form formulas, and for the gradient of logistic regression models. We also discuss the prospects and challenges of engineering efficient verifiable approximation schemes more generally.

Given a directed graph as input, we show how to efficiently find a shortest (directed, simple) cycle on an even number of vertices. As far as we know, no polynomial-time algorithm was previously known for this problem. In fact, finding any even cycle in a directed graph in polynomial time was open for more than two decades until Robertson, Seymour, and Thomas (Ann. of Math. (2) 1999) and, independently, McCuaig (Electron. J. Combin. 2004; announced jointly at STOC 1997) gave an efficiently testable structural characterisation of even-cycle-free directed graphs.

Given as input two $n$-element sets $\mathcal A,\mathcal B\subseteq\{0,1\}^d$ with $d=c\log n\leq(\log n)^2/(\log\log n)^4$ and a target $t\in \{0,1,\ldots,d\}$, we show how to count the number of pairs $(x,y)\in \mathcal A\times \mathcal B$ with integer inner product $\langle x,y \rangle=t$ deterministically, in $n^2/2^{Ω\bigl(\!\sqrt{\log n\log \log n/(c\log^2 c)}\bigr)}$ time. This demonstrates that one can solve this problem in deterministic subquadratic time almost up to $\log^2 n$ dimensions, nearly matching the dimension bound of a subquadratic randomized detection algorithm of Alman and Williams [FOCS 2015]. We also show how to modify their randomized algorithm to count the pairs w.h.p., to obtain a fast randomized algorithm. Our deterministic algorithm builds on a novel technique of reconstructing a function from sum-aggregates by prime residues, which can be seen as an {\em additive} analog of the Chinese Remainder Theorem. As our second contribution, we relate the fine-grained complexity of the task of counting of vector pairs by inner product to the task of computing a zero-one matrix permanent over the integers.

We show that computing the Tutte polynomial of a linear matroid of dimension k on kO(1) points over a field of kO(1) elements requires kΩ(k) time unless the #ETH—a counting extension of the Exponential Time Hypothesis of Impagliazzo and Paturi [CCC 1999] due to Dell et al. [ACM TALG 2014]—is false. This holds also for linear matroids that admit a representation where every point is associated to a vector with at most two nonzero coordinates. Moreover, we also show that the same is true for computing the Tutte polynomial of a binary matroid of dimension k on kO(1) points with at most three nonzero coordinates in each point's vector. These two results stand in sharp contrast to computing the Tutte polynomial of a k-vertex graph (that is, the Tutte polynomial of a graphic matroid of dimension k—which is representable in dimension k over the binary field so that every vector has exactly two nonzero coordinates), which is known to be computable in 2kkO(1) time [Björklund et al., FOCS 2008]. Our lower-bound proofs proceed in three steps: a classic connection due to Crapo and Rota [1970] between the number of tuples of codewords of full support and the Tutte polynomial of the matroid associated with the code; an earlier-established #ETH-hardness of counting the solutions to a bipartite (d, 2)-CSP on n vertices in do(n) time; and new embeddings of such CSP instances as questions about codewords of full support in a linear code. Geometrically, our hardness results also establish that it is #ETH-hard to compute the volume of proper hyperplane chambers in time ko(k) for a given arrangement of hyperplanes through the origin of a finite k-dimensional vector space over a kO(1)-element field. We complement these lower bounds with two algorithm designs to form essentially a complexity dichotomy under #ETH. The first design computes the Tutte polynomial of a linear matroid of dimension k on kO(1) points in kO(k) arithmetic operations in the base field. The second design generalizes the Björklund et al. algorithm from the graphic case and runs in qk+1kO(1) time for linear matroids of dimension k defined over the q-element field by kO(1) points with at most two nonzero coordinates each.

Abstract We derandomize Valiant’s (J ACM 62, Article 13, 2015) subquadratic-time algorithm for finding outlier correlations in binary data. This demonstrates that it is possible to perform a deterministic subquadratic-time similarity join of high dimensionality. Our derandomized algorithm gives deterministic subquadratic scaling essentially for the same parameter range as Valiant’s randomized algorithm, but the precise constants we save over quadratic scaling are more modest. Our main technical tool for derandomization is an explicit family of correlation amplifiers built via a family of zigzag-product expanders by Reingold et al. (Ann Math 155(1):157–187, 2002). We say that a function $$f:\{-1,1\}^d\rightarrow \{-1,1\}^D$$ f : { - 1 , 1 } d → { - 1 , 1 } D is a correlation amplifier with threshold $$0\le \tau \le 1$$ 0 ≤ τ ≤ 1 , error $$\gamma \ge 1$$ γ ≥ 1 , and strength p an even positive integer if for all pairs of vectors $$x,y\in \{-1,1\}^d$$ x , y ∈ { - 1 , 1 } d it holds that (i) $$|\langle x,y\rangle |<\tau d$$ | ⟨ x , y ⟩ | < τ d implies $$|\langle f(x),f(y)\rangle |\le (\tau \gamma )^pD$$ | ⟨ f ( x ) , f ( y ) ⟩ | ≤ ( τ γ ) p D ; and (ii) $$|\langle x,y\rangle |\ge \tau d$$ | ⟨ x , y ⟩ | ≥ τ d implies $$\left (\frac{\langle x,y\rangle }{\gamma d}\right )^pD \le \langle f(x),f(y)\rangle \le \left (\frac{\gamma \langle x,y\rangle }{d}\right )^pD$$ ⟨ x , y ⟩ γ d p D ≤ ⟨ f ( x ) , f ( y ) ⟩ ≤ γ ⟨ x , y ⟩ d p D .

We consider the problem of finding solutions to systems of polynomial equations over a finite field. Lokshtanov et al. [SODA'17] recently obtained the first worst-case algorithms that beat exhaustive search for this problem. In particular for degree-d equations modulo two in n variables, they gave an O^*(2^{(1-1/(5d))n}) time algorithm, and for the special case d=2 they gave an O^*(2^{0.876n}) time algorithm. We modify their approach in a way that improves these running times to O^*(2^{(1-1/(2.7d))n}) and O^*{2^{0.804n}), respectively. In particular, our latter bound - that holds for all systems of quadratic equations modulo 2 - comes close to the O^*(2^{0.792n}) expected time bound of an algorithm empirically found to hold for random equation systems in Bardet et al. [J. Complexity, 2013]. Our improvement involves three observations: 1) The Valiant-Vazirani lemma can be used to reduce the solution-finding problem to that of counting solutions modulo 2. 2) The monomials in the probabilistic polynomials used in this solution-counting modulo 2 have a special form that we exploit to obtain better bounds on their number than in Lokshtanov et al. [SODA'17]. 3) The problem of solution-counting modulo 2 can be "embedded" in a smaller instance of the original problem, which enables us to apply the algorithm as a subroutine to itself.

We study tensor networks as a model of arithmetic computation for evaluating multilinear maps. These capture any algorithm based on low border rank tensor decompositions, such as $O(n^{ω+ε})$ time matrix multiplication, and in addition many other algorithms such as $O(n \log n)$ time discrete Fourier transform and $O^*(2^n)$ time for computing the permanent of a matrix. However tensor networks sometimes yield faster algorithms than those that follow from low-rank decompositions. For instance the fastest known $O(n^{(ω+ε)t})$ time algorithms for counting $3t$-cliques can be implemented with tensor networks, even though the underlying tensor has border rank $n^{3t}$ for all $t \ge 2$. For counting homomorphisms of a general pattern graph $P$ into a host graph on $n$ vertices we obtain an upper bound of $O(n^{(ω+ε)\operatorname{bw}(P)/2})$ where $\operatorname{bw}(P)$ is the branchwidth of $P$. This essentially matches the bound for counting cliques, and yields small improvements over previous algorithms for many choices of $P$. While powerful, the model still has limitations, and we are able to show a number of unconditional lower bounds for various multilinear maps, including: (a) an $Ω(n^{\operatorname{bw}(P)})$ time lower bound for counting homomorphisms from $P$ to an $n$-vertex graph, matching the upper bound if $ω= 2$. In particular for $P$ a $v$-clique this yields an $Ω(n^{\lceil 2v/3 \rceil})$ time lower bound for counting $v$-cliques, and for $P$ a $k$-uniform $v$-hyperclique we obtain an $Ω(n^v)$ time lower bound for $k \ge 3$, ruling out tensor networks as an approach to obtaining non-trivial algorithms for hyperclique counting and the Max-$3$-CSP problem. (b) an $Ω(2^{0.918n})$ time lower bound for the permanent of an $n \times n$ matrix.

We introduce probabilistic extensions of classical deterministic measures of algebraic complexity of a tensor, such as the rank and the border rank. We show that these probabilistic extensions satisfy various natural and algorithmically serendipitous properties, such as submultiplicativity under taking of Kronecker products. Furthermore, the probabilistic extensions enable improvements over their deterministic counterparts for specific tensors of interest, starting from the tensor 〈2, 2, 2〉 that represents 2 × 2 matrix multiplication. While it is well known that the (deterministic) tensor rank and border rank satisfy [V. Strassen, Numer. Math. 13 (1969); J. E. Hopcroft and L. R. Kerr, SIAM J. Appl. Math. 20 (1971); S. Winograd, Linear Algebra Appl. 4 (1971); J. M. Landsberg, J. AMS 19 (2006)], we show that the probabilistic tensor rank and border rank satisfy By submultiplicativity, this leads immediately to novel randomized algorithm designs, such as algorithms for Boolean matrix multiplication as well as detecting and estimating the number of triangles in graphs. Our algorithms are opportunistic in the sense that their worst-case scaling is essentially governed by the probabilistic rank, yet their result is accumulated through independent repetitions, where the partial result can be inspected at each repeat for possible early termination, and each repeat scales according to the rank of the outcome-tensors. For example, representing 〈2, 2, 2〉 probabilistically using an ensemble of tensors of rank 6, we obtain an algorithm that, with high probability, multiplies two 2d × 2d Boolean matrices in operations. This algorithm consists of independent repeats that each run in O(6d) operations and enable inspection of the partial result at each repeat. Analogously, a probabilistic representation of 〈2, 2, 2〉 using tensors of border rank 5 gives an algorithm that runs in operations, consisting of repeats that run in Õ(5d) operations each. Asymptotically, we use Adleman's argument to show that, over the complex field, the support rank exponent ωs of matrix multiplication [H. Cohn and C. Umans, SODA’12] gives the lower bound for probabilistic tensor rank. While this enables an approach to obtaining asymptotically faster algorithm designs for matrix multiplication via the Cohn–Umans inequality , the main motivation for the present paper is to enable an approach towards fast practical algorithms using small probabilistic tensors.

We present two new data structures for computing values of an n-variate polynomial P of degree at most d over a finite field of q elements. Assuming that d divides $$q-1$$ , our first data structure relies on $$(d+1)^{n+2}$$ tabulated values of P to produce the value of P at any of the $$q^n$$ points using $$O(nqd^2)$$ arithmetic operations in the finite field. Assuming that s divides d and d / s divides $$q-1$$ , our second data structure assumes that P satisfies a degree-separability condition and relies on $$(d/s+1)^{n+s}$$ tabulated values to produce the value of P at any point using $$O\left( nq^ssq\right) $$ arithmetic operations. Our data structures are based on generalizing upper-bound constructions due to Mockenhaupt and Tao (Duke Math J 121(1):35–74, 2004), Saraf and Sudan (Anal PDE 1(3):375–379, 2008) and Dvir (Incidence theorems and their applications, 2012. arXiv:1208.5073 ) for Kakeya sets in finite vector spaces from linear to higher-degree polynomial curves. As an application we show that the new data structures enable a faster algorithm for computing integer-valued fermionants, a family of self-reducible polynomial functions introduced by Chandrasekharan and Wiese (Partition functions of strongly correlated electron systems as fermionants, 2011. arXiv:1108.2461v1 ) that captures numerous fundamental algebraic and combinatorial functions such as the determinant, the permanent, the number of Hamiltonian cycles in a directed multigraph, as well as certain partition functions of strongly correlated electron systems in statistical physics. In particular, a corollary of our main theorem for fermionants is that the permanent of an $$m\times m$$ integer matrix with entries bounded in absolute value by a constant can be computed in time $$2^{m-\Omega \left( \sqrt{m/\log \log m}\right) }$$ , improving an earlier algorithm of Björklund (in: Proceedings of the 15th SWAT, vol 17, pp 1–11, 2016) that runs in time $$2^{m-\Omega \left( \sqrt{m/\log m}\right) }$$ .

In the Steiner Tree problem, we are given as input a connected $n$-vertex graph with edge weights in $\{1,2,\ldots,W\}$, and a set of $k$ terminal vertices. Our task is to compute a minimum-weight tree that contains all of the terminals. The main result of the paper is an algorithm solving Steiner Tree in time $\mathcal{O}(7.97^k\cdot n^4\cdot \log{W})$ and using $\mathcal{O}(n^3\cdot \log{nW} \cdot \log k)$ space. This is the first single-exponential time, polynomial space FPT algorithm for the weighted Steiner Tree problem. Whereas our main result seeks to optimize the polynomial dependency in $n$ for both the running time and space usage, it is possible to trade between polynomial dependence in $n$ and the single-exponential dependence in $k$ to obtain faster running time as a function of $k$, but at the cost of increased running time and space usage as a function of $n$. In particular, we show that there exists a polynomial space algorithm for Steiner Tree running in $\mathcal{O}(6.751^kn^{O(1)}\log W)$ time. Finally, by pushing such a trade-off between a polynomial in $n$ and an exponential in $k$ dependencies, we show that for any $\epsilon>0$ there is an $n^{\mathcal{O}(f(\epsilon))}\log W$ space $4^{(1+\epsilon)k}n^{\mathcal{O}(f(\epsilon))}\log W$ time algorithm for Steiner Tree, where $f$ is a computable function depending only on $\epsilon$.

We study the problem of counting the number of occurrences of a given six-vertex pattern graph S in an n-vertex host graph H. We engineer an open-source GPU implementation of a distributed algorithm design of Björklund and Kaski [PODC 2016] where (i) the execution of the algorithm can be delegated [Goldwasser, Kalai, and Rothblum, J. ACM 2015] to produce a noninteractive probabilistically checkable proof of correctness, and (ii) the execution of the algorithm when preparing the proof tolerates a controllable number of adversarial errors. Experiments with NVIDIA Tesla K80 and Tesla P100 Accelerators demonstrate that the framework is practical for inputs of up to 512 vertices, with proof checking being several orders of magnitude more efficient than preparing the proof; however, proof preparation still carries at least one order of magnitude overhead compared with just solving the problem.

We study the problem of counting the isomorphic occurrences of a k-vertex pattern graph P as a subgraph in an n-vertex host graph G. Our specific interest is on algorithms for subgraph counting that are sensitive to the maximum degree Delta of the host graph. Assuming that the pattern graph P is connected and admits a vertex balancer of size b, we present an algorithm that counts the occurrences of P in G in O ((2 Delta-2)^{(k+b)/2} 2^{-b} n/(Delta) k^2 log n) time. We define a balancer as a vertex separator of P that can be represented as an intersection of two equal-size vertex subsets, the union of which is the vertex set of P, and both of which induce connected subgraphs of P. A corollary of our main result is that we can count the number of k-vertex paths in an n-vertex graph in O((2 Delta-2)^{floor[k/2]} n k^2 log n) time, which for all moderately dense graphs with Delta <= n^{1/3} improves on the recent breakthrough work of Curticapean, Dell, and Marx [STOC 2017], who show how to count the isomorphic occurrences of a q-edge pattern graph as a subgraph in an n-vertex host graph in time O(q^q n^{0.17q}) for all large enough q. Another recent result of Brand, Dell, and Husfeldt [STOC 2018] shows that k-vertex paths in a bounded-degree graph can be approximately counted in O(4^kn) time. Our result shows that the exact count can be recovered at least as fast for Delta<10. Our algorithm is based on the principle of inclusion and exclusion, and can be viewed as a sparsity-sensitive version of the "counting in halves"-approach explored by Björklund, Husfeldt, Kaski, and Koivisto [ESA 2009].

Given a vertex-colored graph H and a multiset M of colors as input, the graph motif problem asks us to decide whether H has a connected induced subgraph whose multiset of colors agrees with M. The graph motif problem is NP-complete but known to admit randomized algorithms based on constrained multilinear sieving over GF(2^b) that run in time O(2^kk^2m {M({2^b})}) and with a false-negative probability of at most k/2^{b-1} for a connected m-edge input and a motif of size k. On modern CPU microarchitectures such algorithms have practical edge-linear scalability to inputs with billions of edges for small motif sizes, as demonstrated by Björklund, Kaski, Kowalik, and Lauri [ALENEX'15]. This scalability to large graphs prompts the dual question whether it is possible to scale to large motif sizes. We present a vertex-localized variant of the constrained multilinear sieve that enables us to obtain, in time O(2^kk^2m{M({2^b})}) and for every vertex simultaneously, whether the vertex participates in at least one match with the motif, with a per-vertex probability of at most k/2^{b-1} for a false negative. Furthermore, the algorithm is easily vector-parallelizable for up to 2^k threads, and parallelizable for up to 2^kn threads, where n is the number of vertices in H. Here {M({2^b})} is the time complexity to multiply in GF(2^b). We demonstrate with an open-source implementation that our variant of constrained multilinear sieving can be engineered for vector-parallel microarchitectures to yield hardware utilization that is bound by the available memory bandwidth. Our main engineering contributions are (a) a version of the recurrence for tightly labeled arborescences that can be executed as a sequence of memory-and-arithmetic coalescent parallel workloads on multiple GPUs, and (b) a bit-sliced low-level implementation for arithmetic in characteristic 2 to support (a).

We study the problem of detecting outlier pairs of strongly correlated variables among a collection of n variables with otherwise weak pairwise correlations. After normalization, this task amounts to the geometric task where we are given as input a set of n vectors with unit Euclidean norm and dimension d , and for some constants 0<τ < ρ < 1, we are asked to find all the outlier pairs of vectors whose inner product is at least ρ in absolute value, subject to the promise that all but at most q pairs of vectors have inner product at most τ in absolute value. Improving on an algorithm of Valiant [FOCS 2012; J. ACM 2015], we present a randomized algorithm that for Boolean inputs ({ −1,1}-valued data normalized to unit Euclidean length) runs in time Õ(( n max,{ 1−γ + M (Δ γ ,γ), M (1−γ ,2 Δ γ)} + qdn 2γ ), where 0<γ < 1 is a constant tradeoff parameter and M (μ, ν) is the exponent to multiply an ⌊ n μ ⌋ × ⌊ n ν ⌋ matrix with an ⌊ n ν ⌋ × ⌊ n μ ⌋ matrix and Δ =1/(1−log τ ρ). As corollaries we obtain randomized algorithms that run in time Õ( ( n 2/ω 3−log τ ρ + qdn 2/(1−log τ ρ)3=log ττ ρ ) and in time õ( ( n 4 / 2+α (1−log τ ρ) + qdn 2/α (1−log τ ρ)2+α (1−log τ ρ) >), where 2≤ ω <2.38 is the exponent for square matrix multiplication and 0.3<α ≤ 1 is the exponent for rectangular matrix multiplication. The notation Õ(ṡ) hides polylogarithmic factors in n and d whose degree may depend on ρ and τ. We present further corollaries for the light bulb problem and for learning sparse Boolean functions.

Two sets of 0–1 vectors of fixed length form a uniquely decodeable code pair if their Cartesian product is of the same size as their sumset, where the addition is pointwise over integers. For the size of the sumset of such a pair, van Tilborg has given an upper bound in the general case. Urbanke and Li, and later Ordentlich and Shayevitz, have given better bounds in the unbalanced case, that is, when either of the two sets is sufficiently large. Improvements to the latter bounds are presented.

We are motivated by a tantalizing open question in exact algorithms: can we detect whether an n-vertex directed graph G has a Hamiltonian cycle in time significantly less than 2^n? We present new randomized algorithms that improve upon several previous works: 1. We show that for any constant 0<lambda<1 and prime p we can count the Hamiltonian cycles modulo p^((1-lambda)n/(3p)) in expected time less than c^n for a constant c<2 that depends only on p and lambda. Such an algorithm was previously known only for the case of counting modulo two [Bj\"orklund and Husfeldt, FOCS 2013]. 2. We show that we can detect a Hamiltonian cycle in O^*(3^(n-alpha(G))) time and polynomial space, where alpha(G) is the size of the maximum independent set in G. In particular, this yields an O^*(3^(n/2)) time algorithm for bipartite directed graphs, which is faster than the exponential-space algorithm in [Cygan et al., STOC 2013]. Our algorithms are based on the algebraic combinatorics of "incidence assignments" that we can capture through evaluation of determinants of Laplacian-like matrices, inspired by the Matrix--Tree Theorem for directed graphs. In addition to the novel algorithms for directed Hamiltonicity, we use the Matrix--Tree Theorem to derive simple algebraic algorithms for detecting out-branchings. Specifically, we give an O^*(2^k)-time randomized algorithm for detecting out-branchings with at least k internal vertices, improving upon the algorithms of [Zehavi, ESA 2015] and [Bj\"orklund et al., ICALP 2015]. We also present an algebraic algorithm for the directed k-Leaf problem, based on a non-standard monomial detection problem.

We present two new data structures for computing values of an n-variate polynomial P of degree at most d over a finite field of q elements. Assuming that d divides q-1, our first data structure relies on (d+1)^{n+2} tabulated values of P to produce the value of P at any of the q^n points using O(nqd^2) arithmetic operations in the finite field. Assuming that s divides d and d/s divides q-1, our second data structure assumes that P satisfies a degree-separability condition and relies on (d/s+1)^{n+s} tabulated values to produce the value of P at any point using O(nq^ssq) arithmetic operations. Our data structures are based on generalizing upper-bound constructions due to Mockenhaupt and Tao (2004), Saraf and Sudan (2008), and Dvir (2009) for Kakeya sets in finite vector spaces from linear to higher-degree polynomial curves. As an application we show that the new data structures enable a faster algorithm for computing integer-valued fermionants, a family of self-reducible polynomial functions introduced by Chandrasekharan and Wiese (2011) that captures numerous fundamental algebraic and combinatorial invariants such as the determinant, the permanent, the number of Hamiltonian cycles in a directed multigraph, as well as certain partition functions of strongly correlated electron systems in statistical physics. In particular, a corollary of our main theorem for fermionants is that the permanent of an m-by-m integer matrix with entries bounded in absolute value by a constant can be computed in time 2^{m-Omega(sqrt(m/log log m))}, improving an earlier algorithm of Bjorklund (2016) that runs in time 2^{m-Omega(sqrt(m/log m))}.

This paper presents a technique for symmetry reduction that adaptively assigns a prefix of variables in a system of constraints so that the generated prefix-assignments are pairwise nonisomorphic under the action of the symmetry group of the system. The technique is based on McKay's canonical extension framework (McKay, 1998). Among key features of the technique are (i) adaptability—the prefix sequence can be user-prescribed and truncated for compatibility with the group of symmetries; (ii) parallelizability—prefix-assignments can be processed in parallel independently of each other; (iii) versatility—the method is applicable whenever the group of symmetries can be concisely represented as the automorphism group of a vertex-colored graph; and (iv) implementability—the method can be implemented relying on a canonical labeling map for vertex-colored graphs as the only nontrivial subroutine. To demonstrate the practical applicability of our technique, we have prepared an experimental open-source implementation of the technique and carry out a set of experiments that demonstrate ability to reduce symmetry on hard instances. Furthermore, we demonstrate that the implementation effectively parallelizes to compute clusters with multiple nodes via a message-passing interface.

Vassilevska and Williams (STOC’09) showed how to count simple paths on k vertices and matchings on k /2 edges in an n -vertex graph in time n k /2+ O (1) . In the same year, two different algorithms with the same runtime were given by Koutis and Williams (ICALP’09), and Björklund et al. (ESA’09), via n st /2+ O (1) -time algorithms for counting t -tuples of pairwise disjoint sets drawn from a given family of s -sized subsets of an n -element universe. Shortly afterwards, Alon and Gutner (TALG’10) showed that these problems have Ω( n ⌊ st /2⌋ ) and Ω( n ⌊ k /2⌋ ) lower bounds when counting by color coding. Here, we show that one can do better—we show that the “meet-in-the-middle” exponent st /2 can be beaten and give an algorithm that counts in time n 0.45470382 st + O (1) for t a multiple of three. This implies algorithms for counting occurrences of a fixed subgraph on k vertices and pathwidth p ≪ k in an n -vertex graph in n 0.45470382 k +2 p + O (1) time, improving on the three mentioned algorithms for paths and matchings, and circumventing the color-coding lower bound. We also give improved bounds for counting t -tuples of disjoint s -sets for s = 2,3,4. Our algorithms use fast matrix multiplication. We show an argument that this is necessary to go below the meet-in-the-middle barrier.

Two sets A, B ⊆ {0, 1}nform a Uniquely Decodable Code Pair (UDCP) if every pair a ∈ A, b ∈ B yields a distinct sum a+b, where the addition is over ℤn. We show that every UDCP A, B, with |A| = 2(1−ε)nand |B| = 2βn, satisfies equation. For sufficiently small ε, this bound significantly improves previous bounds by Urbanke and Li [Information Theory Workshop ′98] and Ordentlich and Shayevitz [2014, arXiv:1412.8415], which upper bound β by 0.4921 and 0.4798, respectively, as ε approaches 0.

In this article, the steering committee of the Parameterized Algorithms and Computational Experiments challenge (PACE) reports on the first iteration of the challenge. Where did PACE come from, how did it go, who won, and what's next?

We study a design framework for robust, independently verifiable, and workload-balanced distributed algorithms working on a common input. The framework builds on recent noninteractive Merlin--Arthur proofs of batch evaluation of Williams~[31st IEEE Colloquium on Computational Complexity (CCC'16, May 29-June 1, 2016, Tokyo), to appear] with the basic observation that Merlin's magic is not needed for batch evaluation: mere Knights can prepare the independently verifiable proof, in parallel, and with intrinsic error-correction.

We study the problem of detecting outlier pairs of strongly correlated variables among a collection of n variables with otherwise weak pairwise correlations. After normalization, this task amounts to the geometric task where we are given as input a set of n vectors with unit Euclidean norm and dimension d, and we are asked to find all the outlier pairs of vectors whose inner product is at least ρ in absolute value, subject to the promise that all but at most q pairs of vectors have inner product at most τ in absolute value for some constants 0 < τ < ρ < 1. Improving on an algorithm of G. Valiant [FOCS 2012; J. ACM 2015], we present a randomized algorithm that for Boolean inputs ({–1, 1}-valued data normalized to unit Euclidean length) runs in time where 0 < γ < 1 is a constant tradeoff parameter and M(μ, v) is the exponent to multiply an ⌊nμ⌋ × ⌊nv⌋ matrix with an ⌊nv⌋ × ⌊nμ⌋ matrix and Δ = 1/(1 – logτ ρ). As corollaries we obtain randomized algorithms that run in time and in time where 2 ≤ ω < 2.38 is the exponent for square matrix multiplication and 0.3 < α ≤ 1 is the exponent for rectangular matrix multiplication. We present further corollaries for the light bulb problem and for learning sparse Boolean functions. (The notation Õ(·) hides polylogarithmic factors in n and d whose degree may depend on ρ and τ.)

The SUBSET SUM problem asks whether a given set of n positive integers contains a subset of elements that sum up to a given target t. It is an outstanding open question whether the O^*(2^{n/2})-time algorithm for SUBSET SUM by Horowitz and Sahni [J. ACM 1974] can be beaten in the worst-case setting by a "truly faster", O^*(2^{(0.5-delta)*n})-time algorithm, with some constant delta > 0. Continuing an earlier work [STACS 2015], we study SUBSET SUM parameterized by the maximum bin size beta, defined as the largest number of subsets of the n input integers that yield the same sum. For every epsilon > 0 we give a truly faster algorithm for instances with beta <= 2^{(0.5-epsilon)*n}, as well as instances with beta >= 2^{0.661n}. Consequently, we also obtain a characterization in terms of the popular density parameter n/log_2(t): if all instances of density at least 1.003 admit a truly faster algorithm, then so does every instance. This goes against the current intuition that instances of density 1 are the hardest, and therefore is a step toward answering the open question in the affirmative. Our results stem from a novel combinatorial analysis of mixings of earlier algorithms for SUBSET SUM and a study of an extremal question in additive combinatorics connected to the problem of Uniquely Decodable Code Pairs in information theory.

We introduce a new algebraic sieving technique to detect constrained multilinear monomials in multivariate polynomial generating functions given by an evaluation oracle. The polynomials are assumed to have coefficients from a field of characteristic two. As applications of the technique, we show an $$O^*(2^k)$$ -time polynomial space algorithm for the $$k$$ -sized Graph Motif problem. We also introduce a new optimization variant of the problem, called Closest Graph Motif and solve it within the same time bound. The Closest Graph Motif problem encompasses several previously studied optimization variants, like Maximum Graph Motif, Min-Substitute Graph Motif, and Min-Add Graph Motif. Finally, we provide a piece of evidence that our result might be essentially tight: the existence of an $$O^*((2-\epsilon )^k)$$ -time algorithm for the Graph Motif problem implies an $$O((2-\epsilon ')^n)$$ -time algorithm for Set Cover.

We investigate fast algorithms for changing between the standard basis and an orthogonal basis of idempotents for Möbius algebras of finite lattices. We show that every lattice with v elements, n of which are nonzero and join-irreducible (or, by a dual result, nonzero and meet-irreducible), has arithmetic circuits of size O ( vn ) for computing the zeta transform and its inverse, thus enabling fast multiplication in the Möbius algebra. Furthermore, the circuit construction in fact gives optimal (up to constants) monotone circuits for several lattices of combinatorial and algebraic relevance, such as the lattice of subsets of a finite set, the lattice of set partitions of a finite set, the lattice of vector subspaces of a finite vector space, and the lattice of positive divisors of a positive integer.

In the graph motif problem, we are given as input a vertex-colored graph H (the host graph) and a multiset of colors M (the motif). Our task is to decide whether H has a connected set of vertices whose multiset of colors agrees with M. The graph motif problem is NP-complete but known to admit parameterized algorithms that run in linear time in the size of H. We demonstrate that algorithms based on constrained multilinear sieving are viable in practice, scaling to graphs with hundreds of millions of edges as long as M remains small. Furthermore, our implementation is topology-invariant relative to the host graph H, meaning only the most crude graph parameters (number of edges and number of vertices) suffice in practice to determine the algorithm performance.

In this work, we use algebraic methods for studying distance computation and subgraph detection tasks in the congested clique model. Specifically, we adapt parallel matrix multiplication implementations to the congested clique, obtaining an O(n1-2/ω) round matrix multiplication algorithm, where ω < 2.3728639 is the exponent of matrix multiplication. In conjunction with known techniques from centralised algorithmics, this gives significant improvements over previous best upper bounds in the congested clique model. The highlight results include: triangle and 4-cycle counting in O(n0.158) rounds, improving upon the O(n1/3) triangle counting algorithm of Dolev et al. [DISC 2012], a (1 + o(1))-approximation of all-pairs shortest paths in O(n0.158) rounds, improving upon the ~O (n1/2)-round (2 + o(1))-approximation algorithm of Nanongkai [STOC 2014], and computing the girth in O(n0.158) rounds, which is the first non-trivial solution in this model. In addition, we present a novel constant-round combinatorial algorithm for detecting 4-cycles.

We study the exact time complexity of the Subset Sum problem. Our focus is on instances that lack additive structure in the sense that the sums one can form from the subsets of the given integers are not strongly concentrated on any particular integer value. We present a randomized algorithm that runs in O(2^0.3399nB^4) time on instances with the property that no value can arise as a sum of more than B different subsets of the n given integers.

Vassilevska and Williams (STOC 2009) showed how to count simple paths on k vertices and matchings on k/2 edges in an n-vertex graph in time nk/2+O(1). In the same year, two different algorithms with the same runtime were given by Koutis and Williams (ICALP 2009), and Björklund et al. (ESA 2009), via nst/2+O(1)-time algorithms for counting t-tuples of pairwise disjoint sets drawn from a given family of s-sized subsets of an n-element universe. Shortly afterwards, Alon and Gutner (TALG 2010) showed that these problems have Ω(n⌊st/2⌋) and Ω(n⌊k/2⌋) lower bounds when counting by color coding. Here we show that one can do better, namely, we show that the “meet-in-the-middle” exponent st/2 can be beaten and give an algorithm that counts in time n0.4547st+O(1) for t a multiple of three. This implies algorithms for counting occurrences of a fixed subgraph on k vertices and pathwidth p ≪ k in an n-vertex graph in n0.4547k+2p+O(1) time, improving on the three mentioned algorithms for paths and matchings, and circumventing the color-coding lower bound.

We show an O^*(2^k)-time polynomial space algorithm for the k-sized Graph Motif problem. We also introduce a new optimization variant of the problem, called Closest Graph Motif and solve it within the same time bound. The Closest Graph Motif problem encompasses several previously studied optimization variants, like Maximum Graph Motif, Min-Substitute, and Min-Add. Moreover, we provide a piece of evidence that our result might be essentially tight: the existence of an O^*((2-epsilon)^k)-time algorithm for the Graph Motif problem implies an ((2-epsilon')^n)-time algorithm for Set Cover.

We investigate fast algorithms for changing between the standard basis and an orthogonal basis of idempotents for Möbius algebras of finite lattices. We show that every lattice with v elements, n of which are nonzero and join-irreducible (or, by a dual result, nonzero and meet-irreducible), has arithmetic circuits of size O(vn) for computing the zeta transform and its inverse, thus enabling fast multiplication in the Möbius algebra. Furthermore, the circuit construction in fact gives optimal (up to constants) circuits for a number of lattices of combinatorial and algebraic relevance, such as the lattice of subsets of a finite set, the lattice of set partitions of a finite set, the lattice of vector subspaces of a finite vector space, and the lattice of positive divisors of a positive integer.

We show that the traveling salesman problem in bounded-degree graphs can be solved in time O ((2-ϵ) n ), where ϵ > 0 depends only on the degree bound but not on the number of cities, n . The algorithm is a variant of the classical dynamic programming solution due to Bellman, and, independently, Held and Karp. In the case of bounded integer weights on the edges, we also give a polynomial-space algorithm with running time O ((2-ϵ) n ) on bounded-degree graphs. In addition, we present an analogous analysis of Ryser's algorithm for the permanent of matrices with a bounded number of nonzero entries in each column.

A binary matrix is fully nested if its columns form a chain of subsets; that is, any two columns are ordered by the subset relation, where we view each column as a subset of the rows indicated by the 1-entries. A binary matrix is k-nested if its columns can be partitioned into k pairwise disjoint blocks, each of which is fully nested. Such nested patterns are encountered, for example, in presence/absence patterns of species in ecological data. We study the automated discovery of k-nestedness on synthetic data and real ecological data. First, we show that k-nestedness can be efficiently discovered in a noise-free setting using a polynomial-time algorithm. Second, we show that it is NP-hard to find a k-nested matrix that minimizes the Hamming distance to a given dataset. Thus, it is likely that in the presence of noise no efficient algorithm exists for discovering k-nestedness in the general case. Third, we develop and evaluate multiple heuristic algorithms for discovering k-nestedness on noisy synthetic data. The methods based on a combination of singular value decomposition and k-means++ give the best performance in terms of structure discovery and noise tolerance. Fourth, we develop an MDL-based model selection technique for assessing nestedness, and discover k-nested structure in (a) paleontological data, and (b) geographical occurrence data for mammal species in Europe.

Time series are a class of data whose complexity and rich structure make it difficult for data mining tools to extract meaningful patterns from them, and in particular to prune away the false positive patterns. Wavelet-based methods have recently become the preferred way for significance testing of time series and time series collections, but these methods are still often based on fairly ad hoc bootstrapping techniques in the wavelet domain without a disciplined null model analysis. We propose a new well-grounded null model for time series collections that also sets minimum requirements for realistic resampling methods. We compare it to the null models of common resampling methods and introduce a new randomization method that is compatible with the proposed null model. We conduct experiments on real and synthetic datasets to compare the behavior of the various methods and reflect the results to the differences in their null models. Compared with the other methods, our experiments suggest that the proposed method gives fewer Type I and Type II errors across a range of statistics.

Clustering is one of the basic operations in data analysis, and the cluster structure of a dataset often has a marked effect on observed patterns in data. Testing whether a data mining result is implied by the cluster structure can give substantial information on the formation of the dataset. We propose a new method for empirically testing the statistical significance of patterns in real-valued data in relation to the cluster structure. The method relies on principal component analysis and is based on the general idea of decomposing the data for the purpose of isolating the null model. We evaluate the performance of the method and the information it provides on various real datasets. Our results show that the proposed method is robust and provides nontrivial information about the origin of patterns in data, such as the source of classification accuracy and the observed correlations between attributes.

We study the stable marriage problem in a distributed setting. The communication network is a bipartite graph, with men on one side and women on the other. Acceptable partners are connected by edges, and each participant has chosen a linear order on the adjacent nodes, indicating the matching preferences.

In a max-min LP, the objective is to maximise ω subject to Ax ≤ 1, Cx ≥ ω1, and x ≥ 0 for nonnegative matrices A and C. We present a local algorithm (constant-time distributed algorithm) for approximating max-min LPs. The approximation ratio of our algorithm is the best possible for any local algorithm; there is a matching unconditional lower bound.

The deletion–contraction algorithm is perhapsthe most popular method for computing a host of fundamental graph invariants such as the chromatic, flow, and reliability polynomials in graph theory, the Jones polynomial of an alternating link in knot theory, and the partition functions of the models of Ising, Potts, and Fortuin–Kasteleyn in statistical physics. Prior to this work, deletion–contraction was also the fastest known general-purpose algorithm for these invariants, running in time roughly proportional to the number of spanning trees in the input graph.Here, we give a substantially faster algorithm that computes the Tutte polynomial—and hence, all the aforementioned invariants and more—of an arbitrary graph in time within a polynomial factor of the number of connected vertex sets. The algorithm actually evaluates a multivariate generalization of the Tutte polynomial by making use of an identity due to Fortuin and Kasteleyn. We also provide a polynomial-space variant of the algorithm and give an analogous result for Chung and Graham's cover polynomial.

A local algorithm is a distributed algorithm where each node must operate solely based on the information that was available at system startup within a constant-size neighbourhood of the node. We study the applicability of local algorithms to max-min LPs where the objective is to maximise minkSigmav CkvXv subject to SigmavalphaivXv les 1 far each i and Xv ges 0 far each v. Here ckvges 0, and the support sets Vi= {v : alphaiv> 0}, Vk= {v : ckv> 0}, Iv= {i: alphaiv> 0} and Kv= {k : Ckv> 0} have bounded size. In the distributed setting, each agent v is responsible for choosing the value of Xv, and the communication network is a hypergraph H where the sets Vkand Viconstitute the hyperedges. We present inapproximability results for a wide range of structural assumptions; for example, even if |Vi| and |Vk| are bounded by some constants larger than 2, there is no local approximation scheme. To contrast the negative results, we present a local approximation algorithm which achieves good approximation ratios if we can bound the relative growth of the vertex neighbourhoods in H.

We study ways to expedite Yates's algorithm for computing the zeta and Moebius transforms of a function defined on the subset lattice. We develop a trimmed variant of Moebius inversion that proceeds point by point, finishing the calculation at a subset before considering its supersets. For an $n$-element universe $U$ and a family $\scr F$ of its subsets, trimmed Moebius inversion allows us to compute the number of packings, coverings, and partitions of $U$ with $k$ sets from $\scr F$ in time within a polynomial factor (in $n$) of the number of supersets of the members of $\scr F$. Relying on an intersection theorem of Chung et al. (1986) to bound the sizes of set families, we apply these ideas to well-studied combinatorial optimisation problems on graphs of maximum degree $Δ$. In particular, we show how to compute the Domatic Number in time within a polynomial factor of $(2^{Δ+1-2)^{n/(Δ+1)$ and the Chromatic Number in time within a polynomial factor of $(2^{Δ+1-Δ-1)^{n/(Δ+1)$. For any constant $Δ$, these bounds are $O\bigl((2-ε)^n\bigr)$ for $ε>0$ independent of the number of vertices $n$.

The problem of canonically labeling a graph is studied. Within the general framework of backtracking algorithms based on individualization and refinement, data structures, subroutines, and pruning heuristics especially for fast handling of large and sparse graphs are developed. Experiments indicate that the algorithm implementation in most cases clearly outperforms existing state-of-the-art tools.

We present a fast algorithm for the subset convolution problem:given functions f and g defined on the lattice of subsets of ann-element set n, compute their subset convolution f*g, defined for S⊆ N by [ (f * g)(S) = [T ⊆ S] f(T) g(S/T),,]where addition and multiplication is carried out in an arbitrary ring. Via Möbius transform and inversion, our algorithm evaluates the subset convolution in O(n2 2n) additions and multiplications, substanti y improving upon the straightforward O(3n) algorithm. Specifically, if the input functions have aninteger range [-M,-M+1,...,M], their subset convolution over the ordinary sum--product ring can be computed in Õ(2n log M) time; the notation Õ suppresses polylogarithmic factors.Furthermore, using a standard embedding technique we can compute the subset convolution over the max--sum or min--sum semiring in Õ(2n M) time.

We consider the problem of maximizing the lifetime of a given multicast connection in a wireless network of energy-constrained (e.g., battery-operated) nodes, by choosing ideal transmission power levels for the nodes relaying the connection. We distinguish between two basic operating modes: In a static power assignment, the power levels of the nodes are set at the beginning and remain unchanged until the nodes are depleted of energy. In a dynamic power schedule, the powers can be adjusted during operation. We show that while lifetime-maximizing static power assignments can be found in polynomial time, for dynamic schedules the problem becomes NP-hard. We introduce two approximation heuristics for the dynamic case, and experimentally verify that the lifetime of a dynamically adjusted multicast connection can be made several times longer than what can be achieved by the best possible static assignment.

We develop an algorithm framework for isomorph-free exhaustive generation of designs admitting a group of automorphisms from a prescribed collection of pairwise nonconjugate groups, where each prescribed group has a large index relative to its normalizer in the isomorphism-inducing group. We demonstrate the practicality of the framework by producing a complete classification of the Steiner triple systems of order $21$ admitting a nontrivial automorphism group. The number of such pairwise nonisomorphic designs is $62336617$, where $958$ of the designs are anti-Pasch. We also develop consistency checking methodology for gaining confidence in the correct operation of the algorithm implementation.