Holger Dell

Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses

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 ε we show that if satisfiability for n-variable d-CNF formulas has a protocol of cost O(nd-ε) 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 ε = 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(k2-ε) 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.

On Problems as Hard as CNF-SAT

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 epsilon

Satisfiability Allows No Nontrivial Sparsification unless the Polynomial-Time Hierarchy Collapses

Consider the following two-player communication process to decide a language <i>L</i>: The first player holds the entire input <i>x</i> but is polynomially bounded; the second player is computationally unbounded but does not know any part of <i>x</i>; their goal is to decide cooperatively whether <i>x</i> belongs to <i>L</i> 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 <i>d</i> ≥ 3 and positive real <i>ε</i>, we show that, if satisfiability for <i>n</i>-variable <i>d</i>-CNF formulas has a protocol of cost <i>O</i>(<i>nd</i> − <i>ε</i>), 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 <i>ε</i> = 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 <i>n</i>-vertex <i>d</i>-uniform hypergraphs, this statement holds for any integer <i>d</i> ≥ 2. The case <i>d</i> = 2 implies that no NP-hard vertex deletion problem based on a graph property that is inherited by subgraphs can have kernels consisting of <i>O</i>(<i>k</i>2 − <i>ε</i>) edges unless coNP is in NP/poly, where <i>k</i> denotes the size of the deletion set. Kernels consisting of <i>O</i>(<i>k</i>2) edges are known for several problems in the class, including vertex cover, feedback vertex set, and bounded-degree deletion.

Complexity of the cover polynomial

The cover polynomial introduced by Chung and Graham is a two-variate graph polynomial for directed graphs. It counts the (weighted) number of ways to cover a graph with disjoint directed cycles and paths, it is an interpolation between determinant and permanent, and it is believed to be a directed analogue of the Tutte polynomial. Jaeger, Vertigan, and Welsh showed that the Tutte polynomial is #Phard to evaluate at all but a few special points and curves. It turns out that the same holds for the cover polynomial: We prove that, in almost the whole plane, the problem of evaluating the cover polynomial is #Phard under polynomial-time Turing reductions, while only three points are easy. Our construction uses a gadget which is easier to analyze and more general than the XOR-gadget used by Valiant in his proof that the permanent is #P-complete.

Exponential Time Complexity of the Permanent and the Tutte Polynomial

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

Exponential time complexity of the permanent and the Tutte polynomial

The Exponential Time Hypothesis (ETH) says that deciding the satisfiability of n-variable 3-CNF formulas requires time exp(Ω(n)). We relax this hypothesis by introducing its counting version #;ETH, namely that every algorithm that counts the satisfying assignments requires time exp(Ω(n)). We transfer the sparsification lemma for d-CNF formulas to the counting setting, which makes #ETH robust. Under this hypothesis, we show lower bounds for well-studied #P-hard problems: Computing the permanent of an n×n matrix with m nonzero entries requires time exp(Ω(m)). Restricted to 01-matrices, the bound is exp(Ω(m/logm)). Computing the Tutte polynomial of a multigraph with n vertices and m edges requires time exp(Ω(n)) at points (x, y) with (x - 1)(y - 1) ≠ = 1 and y ∉ {0,±1}. At points (x, 0) with x ∉ {0,±1} it requires time exp(Ω(n)), and if x = -2,-3, ..., it requires time exp(Ω(m)). For simple graphs, the bound is exp(Ω(m/log3 m)).

Homomorphisms are a good basis for counting small subgraphs

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. We use the framework of graph motif parameters to obtain faster algorithms for counting subgraph copies of fixed graphs H in host graphs G. More precisely, for graphs H on k edges, we show how to count subgraph copies of H in time kO(k)· n0.174k + o(k) by a surprisingly simple algorithm. This improves upon previously known running times, such as O(n0.91k + c) time for k-edge matchings or O(n0.46k + 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 ε 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: For vertex-colored graphs H and G, where H is from a fixed class of graphs, we want to count color-preserving H-copies in G. We show that this problem is either polynomial-time solvable or FPT or #W[1]-hard, and that the FPT cases indeed need FPT time under reasonable assumptions.

Complexity of the Bollobás-Riordan Polynomial

The coloured Tutte polynomial by Bollobas and Riordan is, as a generalization of the Tutte polynomial, the most general graph polynomial for coloured graphs that satisfies certain contraction-deletion identities. Jaeger, Vertigan, and Welsh showed that the classical Tutte polynomial is #P-hard to evaluate almost everywhere by establishing reductions along curves and lines.

The PACE 2017 Parameterized Algorithms and Computational Experiments Challenge: The Second Iteration

In this article, the Program Committee of the Second Parameterized Algorithms and Computational Experiments challenge (PACE 2017) reports on the second iteration of the PACE challenge. Track A featured the Treewidth problem and Track B the Minimum Fill-In problem. Over 44 participants on 17 teams from 11 countries submitted their implementations to the competition.

Complexity of the Bollobás–Riordan Polynomial. Exceptional Points and Uniform Reductions

The coloured Tutte polynomial by Bollobás and Riordan is, as a generalization of the Tutte polynomial, the most general graph polynomial for coloured graphs that satisfies certain contraction-deletion identities. Jaeger, Vertigan, and Welsh showed that the classical Tutte polynomial is #P-hard to evaluate almost everywhere by establishing reductions along curves and lines.We establish a similar result for the coloured Tutte polynomial on integral domains. To capture the algebraic flavour and the uniformity inherent in this type of result, we introduce a new kind of reductions, uniform algebraic reductions, that are well-suited to investigate the evaluation complexity of graph polynomials. Our main result identifies a small, algebraic set of exceptional points and says that the evaluation problem of the coloured Tutte is equivalent for all non-exceptional points, under polynomial-time uniform algebraic reductions. On the way we obtain a self-contained proof for the difficult evaluations of the classical Tutte polynomial.