Consider the following two-player communication process to decide a language L: The first player holds the entire input x but is polynomially bounded; the second player is computationally unbounded but does not know any part of x; their goal is to decide cooperatively whether x belongs to L at small cost, where the cost measure is the number of bits of communication from the first player to the second player.For any integer d ≥ 3 and positive real , we show that, if satisfiability for n-variable d-CNF formulas has a protocol of cost O(n d− ), then coNP is in NP/poly, which implies that the polynomial-time hierarchy collapses to its third level. The result even holds when the first player is conondeterministic, and is tight as there exists a trivial protocol for = 0. Under the hypothesis that coNP is not in NP/poly, our result implies tight lower bounds for parameters of interest in several areas, namely sparsification, kernelization in parameterized complexity, lossy compression, and probabilistically checkable proofs.By reduction, similar results hold for other NP-complete problems. For the vertex cover problem on nvertex d-uniform hypergraphs, this statement holds for any integer d ≥ 2. The case d = 2 implies that no NP-hard vertex deletion problem based on a graph property that is inherited by subgraphs can have kernels consisting of O(k 2− ) edges unless coNP is in NP/poly, where k denotes the size of the deletion set. Kernels consisting of O(k 2 ) edges are known for several problems in the class, including vertex cover, feedback vertex set, and bounded-degree deletion. ACM Reference Format:Dell, H. and van Melkebeek, D. 2014. Satisfiability allows no nontrivial sparsification unless the polynomialtime hierarchy collapses.

We introduce graph motif parameters, a class of graph parameters that depend only on the frequencies of constant-size induced subgraphs. Classical works by Lovász show that many interesting quantities have this form, including, for fixed graphs H, the number of H-copies (induced or not) in an input graph G, and the number of homomorphisms from H to G.Using the framework of graph motif parameters, we obtain faster algorithms for counting subgraph copies of fixed graphs H in host graphs G: For graphs H on k edges, we show how to count subgraph copies ofby a surprisingly simple algorithm. This improves upon previously known running times, such as O(n 0.91k+c ) time for k-edge matchings or O(n 0.46k+c ) time for k-cycles. Furthermore, we prove a general complexity dichotomy for evaluating graph motif parameters: Given a class C of such parameters, we consider the problem of evaluating f ∈ C on input graphs G, parameterized by the number of induced subgraphs that f depends upon. For every recursively enumerable class C, we prove the above problem to be either FPT or #W[1]-hard, with an explicit dichotomy criterion. This allows us to recover known dichotomies for counting subgraphs, induced subgraphs, and homomorphisms in a uniform and simplified way, together with improved lower bounds.Finally, we extend graph motif parameters to colored subgraphs and prove a complexity trichotomy:

The field of exact exponential time algorithms for NP-hard problems has thrived over the last decade. While exhaustive search remains asymptotically the fastest known algorithm for some basic problems, difficult and non-trivial exponential time algorithms have been found for a myriad of problems, including Graph Coloring, Hamiltonian Path, Dominating Set and 3-CNF-Sat. In some instances, improving these algorithms further seems to be out of reach. The CNF-Sat problem is the canonical example of a problem for which the trivial exhaustive search algorithm runs in time O(2 n ), where n is the number of variables in the input formula. While there exist non-trivial algorithms for CNF-Sat that run in time o(2 n ), no algorithm was able to improve the growth rate 2 to a smaller constant, and hence it is natural to conjecture that 2 is the optimal growth rate. The strong exponential time hypothesis (SETH) by Impagliazzo and Paturi [JCSS 2001] goes a little bit further and asserts that, for every < 1, there is a (large) integer k such that k-CNF-Sat cannot be computed in time 2 n . In this paper, we show that, for every < 1, the problems Hitting Set, Set Splitting, and NAE-Sat cannot be computed in time O(2 n ) unless SETH fails. Here n is the number of elements or variables in the input. For these problems, we actually get an equivalence to SETH in a certain sense. We conjecture that SETH implies a similar statement for Set Cover, and prove that, under this assumption, the fastest known algorithms for Steiner Tree, Connected Vertex Cover, Set Partitioning, and the pseudo-polynomial time algorithm for Subset Sum cannot be significantly improved. Finally, we justify our assumption about the hardness of Set Cover by showing that the parity of the number of solutions to Set Cover cannot be computed in time O(2 n ) for any < 1 unless SETH fails.

Consider the following two-player communication process to decide a language L: The first player holds the entire input x but is polynomially bounded; the second player is computationally unbounded but does not know any part of x; their goal is to cooperatively decide whether x belongs to L at small cost, where the cost measure is the number of bits of communication from the first player to the second player.For any integer d ≥ 3 and positive real ǫ we show that if satisfiability for n-variable d-CNF formulas has a protocol of cost O(n d−ǫ ) then coNP is in NP/poly, which implies that the polynomial-time hierarchy collapses to its third level. The result even holds when the first player is conondeterministic, and is tight as there exists a trivial protocol for ǫ = 0. Under the hypothesis that coNP is not in NP/poly, our result implies tight lower bounds for parameters of interest in several areas, namely sparsification, kernelization in parameterized complexity, lossy compression, and probabilistically checkable proofs.By reduction, similar results hold for other NP-complete problems. For the vertex cover problem on n-vertex d-uniform hypergraphs, the above statement holds for any integer d ≥ 2. The case d = 2 implies that no NP-hard vertex deletion problem based on a graph property that is inherited by subgraphs can have kernels consisting of O(k 2−ǫ ) edges unless coNP is in NP/poly, where k denotes the size of the deletion set. Kernels consisting of O(k 2 ) edges are known for several problems in the class, including vertex cover, feedback vertex set, and bounded-degree deletion.

The Strong Exponential Time Hypothesis and the OV-conjecture are two popular hardness assumptions used to prove a plethora of lower bounds, especially in the realm of polynomialtime algorithms. The OV-conjecture in moderate dimension states there is no ε > 0 for which an O(N 2−ε ) poly(D) time algorithm can decide whether there is a pair of orthogonal vectors in a given set of size N that contains D-dimensional binary vectors.We strengthen the evidence for these hardness assumptions. In particular, we show that if the OV-conjecture fails, then two problems for which we are far from obtaining even tiny improvements over exhaustive search would have surprisingly fast algorithms. If the OV conjecture is false, then there is a fixed ε > 0 such that: ACM Subject Classification Theory of computation → Problems, reductions and completenessMore Consequences of Falsifying SETH and the Orthogonal Vectors Conjecture decide the satisfiability of bounded-width CNF formulas. SETH is used in the study of exact and fixed parameter tractable algorithms, see e.g [23,46] or the book by Cygan et al. [24]. In this area, it implies, among other things, tight lower bounds for problems on graphs that have small treewidth or pathwidth [41,26,25].Closely related to SETH, the orthogonal vectors problem (OV) is, given two sets A and B of N vectors from {0, 1} D , to decide whether there are vectors a ∈ A and b ∈ B such that a and b are orthogonal in Z D . If D ≤ O(N 0.3 ) holds, the problem can be solved in timeÕ(N 2 ) using an algorithm based on fast rectangular matrix multiplication (see e.g. [31]). SETH implies [54] that this algorithm is essentially as fast as possible; in particular, SETH implies the following hardness conjecture, which was given its name by Gao et al. [32].Conjecture 1.1 (Moderate-dimension OV Conjecture). There are no reals ε, δ > 0 such that OV for D = N δ can be solved 1 in time O(N 2−ε ).The moderate-dimension OV conjecture is used to study the fine-grained complexity of problems in P, for which it has remarkably strong and diverse implications. If the conjecture is true, then dozens of important problems from all across computer science exhibit running time lower bounds that match existing upper bounds up to subpolynomial factors. These include pattern matching and other problems in bioinformatics [7, 10, 40, 1], graph algorithms [47,6,32], computational geometry [16], formal languages [11,18], time-series analysis [2,19], and even economics [42] (see [58] for a more comprehensive list).Gao et al.[32] also named the low-dimension OV conjecture, which asserts that OV does not have subquadratic algorithms whenever D = ω(log N ) holds. The low-dimension implies the moderate-dimension variant of the OV conjecture, and both are implied by SETH [54]. Recent results on the hardness of approximation problems, such as Maximum Inner Product [5], rely on the stronger conjecture (perhaps also [12,14]). However, for the vast majority of OV-based hardness results, reducing the dimension only affects lower-order terms in the lo...

Kernelization algorithms are polynomial-time reductions from a problem to itself that guarantee their output to have a size not exceeding some bound. We make rst progress in that direction by showing nontrivial kernels with O(k 2.5 ) edges for the problem of nding k vertex-disjoint paths of three edges each. This does not quite match the best lower bound of O(k 2− ) that we can prove.Most of our lower bound proofs follow a general scheme that we discover: To exclude kernels of size O(k d− ) for a problem in d-uniform hypergraphs, one should reduce from a carefully chosen d-partite problem that is still NP-hard. As an illustration, we apply this scheme to the vertex cover problem, which allows us to replace the number-theoretical construction by Dell and Van Melkebeek [STOC 2010] with shorter elementary arguments.

We show conditional lower bounds for well-studied #P-hard problems:• The number of satisfying assignments of a 2-CNF formula with n variables cannot be computed in time exp(o(n)), and the same is true for computing the number of all independent sets in an n-vertex graph.• The permanent of an n × n matrix with entries 0 and 1 cannot be computed in time exp(o(n)).• The Tutte polynomial of an n-vertex multigraph cannot be computed in time exp(o(n)) at most evaluation points (x, y) in the case of multigraphs, and it cannot be computed in time exp(o(n/ poly log n)) in the case of simple graphs.Our lower bounds are relative to (variants of) the Exponential Time Hypothesis (ETH), which says that the satisfiability of n-variable 3-CNF formulas cannot be decided in time exp(o(n)). We relax this hypothesis by introducing its counting version #ETH, namely that the satisfying assignments cannot be counted in time exp(o(n)). In order to use #ETH for our lower bounds, we transfer the sparsification lemma for d-CNF formulas to the counting setting.

The field of exact exponential time algorithms for NP-hard problems has thrived over the last decade. While exhaustive search remains asymptotically the fastest known algorithm for some basic problems, difficult and non-trivial exponential time algorithms have been found for a myriad of problems, including Graph Coloring, Hamiltonian Path, Dominating Set and 3-CNF-Sat. In some instances, improving these algorithms further seems to be out of reach. The CNF-Sat problem is the canonical example of a problem for which the trivial exhaustive search algorithm runs in time O(2 n ), where n is the number of variables in the input formula. While there exist non-trivial algorithms for CNF-Sat that run in time o(2 n ), no algorithm was able to improve the growth rate 2 to a smaller constant, and hence it is natural to conjecture that 2 is the optimal growth rate. The strong exponential time hypothesis (SETH) by Impagliazzo and Paturi [JCSS 2001] goes a little bit further and asserts that, for every < 1, there is a (large) integer k such that k-CNF-Sat cannot be computed in time 2 n . In this paper, we show that, for every < 1, the problems Hitting Set, Set Splitting, and NAE-Sat cannot be computed in time O(2 n ) unless SETH fails. Here n is the number of elements or variables in the input. For these problems, we actually get an equivalence to SETH in a certain sense. We conjecture that SETH implies a similar statement for Set Cover, and prove that, under this assumption, the fastest known algorithms for Steiner Tree, Connected Vertex Cover, Set Partitioning, and the pseudo-polynomial time algorithm for Subset Sum cannot be significantly improved. Finally, we justify our assumption about the hardness of Set Cover by showing that the parity of the number of solutions to Set Cover cannot be computed in time O(2 n ) for any < 1 unless SETH fails.