Marc Thurley

sharpSAT: counting models with advanced component caching and implicit BCP

We introduce sharpSAT, a new #SAT solver that is based on the well known DPLL algorithm and techniques from SAT and #SAT solvers. Most importantly, we introduce an entirely new approach of coding components, which reduces the cache size by at least one order of magnitude, and a new cache management scheme. Furthermore, we apply a well known look ahead based on BCP in a manner that is well suited for #SAT solving. We show that these techniques are highly beneficial, especially on large structured instances, such that our solver performs significantly better than other #SAT solvers.

Clause-learning algorithms with many restarts and bounded-width resolution

We offer a new understanding of some aspects of practical SAT-solvers that are based on DPLL with unit-clause propagation, clause-learning, and restarts. We do so by analyzing a concrete algorithm which we claim is faithful to what practical solvers do. In particular, before making any new decision or restart, the solver repeatedly applies the unit-resolution rule until saturation, and leaves no component to the mercy of non-determinism except for some internal randomness. We prove the perhaps surprising fact that, although the solver is not explicitly designed for it, with high probability it ends up behaving as width-k resolution after no more than O(n2k+2) conflicts and restarts, where n is the number of variables. In other words, width-k resolution can be thought of as O(n2k+2) restarts of the unit-resolution rule with learning.

Understanding the Complexity of Induced Subgraph Isomorphisms

We study left-hand side restrictions of the induced subgraph isomorphismproblem: Fixing a class , for given graphs G and arbitrary Hwe ask for induced subgraphs of Hisomorphic to G. For the homomorphism problem this kind of restriction has been studied by Grohe and Dalmau, Kolaitis and Vardi for the decision problem and by Dalmau and Jonsson for its counting variant. We give a dichotomy result for both variants of the induced subgraph isomorphism problem. Under some assumption from parameterized complexity theory, these problems are solvable in polynomial time if and only if contains no arbitrarily large graphs. All classifications are given by means of parameterized complexity. The results are presented for arbitrary structures of bounded arity which implies, for example, analogous results for directed graphs. Furthermore, we show that no such dichotomy is possible in the sense of classical complexity. That is, if there are classes such that the induced subgraph isomorphism problem on is neither in nor -complete. This argument may be of independent interest, because it is applicable to various parameterized problems.

Clause-Learning Algorithms with Many Restarts and Bounded-Width Resolution

We offer a new understanding of some aspects of practical SAT-solvers that are based on DPLL with unit-clause propagation, clause-learning, and restarts. On the theoretical side, we do so by analyzing a concrete algorithm which we claim is faithful to what practical solvers do. In particular, before making any new decision or restart, the solver repeatedly applies the unit-resolution rule until saturation, and leaves no component to the mercy of non-determinism except for some internal randomness. We prove the perhaps surprising fact that, although the solver is not explicitely designed for it, it ends up behaving as width-k resolution after no more than n 2k + 1 conflicts and restarts, where n is the number of variables. In other words, width-k resolution can be thought as n 2k + 1 restarts of the unit-resolution rule with learning. On the experimental side, we give evidence for the claim that this theoretical result describes real world solvers. We do so by running some of the most prominent solvers on some CNF formulas that we designed to have resolution refutations of width k . It turns out that the upper bound of the theoretical result holds for these solvers and that the true performance appears to be not very far from it.

An Approximation Algorithm for #k-SAT

We present a simple randomized algorithm that approximates the number of satisfying assignments of Boolean formulas in conjunctive normal form. To the best of our knowledge this is the first algorithm which approximates #k-SAT for any k 3 within a running time that is not only non-trivial, but also significantly better than that of the currently fastest exact algorithms for the problem. More precisely, our algorithm is a randomized approximation scheme whose running time depends polynomially on the error tolerance and is mildly exponential in the number n of variables of the input formula. For example, even stipulating sub-exponentially small error tolerance, the number of solutions to 3-CNF input formulas can be approximated in time

Kernelizations for parameterized counting problems

Kernelizations are an important tool in designing fixed parameter algorithms for parameterized decision problems. We introduce an analogous notion for counting problems, to wit, counting kernelizations which turn out to be equivalent to the fixed parameter tractability of counting problems. Furthermore, we study the application of well-known kernelization techniques to counting problems. Among these are the Buss Kernelization and the Crown Rule Reduction for the vertex cover problem. Furthermore, we show how to adapt kernelizations for the hitting set problem on hypergraphs with hyperedges of bounded cardinality and the unique hitting set problem to their counting analogs.