Maria Luisa Bonet

Solving (Weighted) Partial MaxSAT through Satisfiability Testing

Recently, Fu and Malik described an unweighted Partial MaxSAT solver based on successive calls to a SAT solver. At the k th iteration the SAT solver tries to certify that there exist an assignment that satisfies all but k clauses. Later Marques-Silva and Planes implemented and extended these ideas. In this paper we present and implement two Partial MaxSAT solvers and the weighted variant of one of them. Both are based on Fu and Malik ideas. We prove the correctness of our algorithm and compare our solver with other (Weighted) MaxSAT and (Weighted) Partial MaxSAT solvers.

On Interpolation and Automatization for Frege Systems

The interpolation method has been one of the main tools for proving lower bounds for propositional proof systems. Loosely speaking, if one can prove that a particular proof system has the feasible interpolation property, then a generic reduction can (usually) be applied to prove lower bounds for the proof system, sometimes assuming a (usually modest) complexity-theoretic assumption. In this paper, we show that this method cannot be used to obtain lower bounds for Frege systems, or even for TC0-Frege systems. More specifically, we show that unless factoring (of Blum integers) is feasible, neither Frege nor TC0-Frege has the feasible interpolation property. In order to carry out our argument, we show how to carry out proofs of many elementary axioms/theorems of arithmetic in polynomial-sized TC0-Frege. As a corollary, we obtain that TC0-Frege, as well as any proof system that polynomially simulates it, is not automatizable (under the assumption that factoring of Blum integers is hard). We also show under the same hardness assumption that the k-provability problem for Frege systems is hard.

SAT-based MaxSAT algorithms

Many industrial optimization problems can be translated to MaxSAT. Although the general problem is NP hard, like SAT, many practical problems may be solved using modern MaxSAT solvers. In this paper we present several algorithms specially designed to deal with industrial or real problems. All of them are based on the idea of solving MaxSAT through successive calls to a SAT solver. We show that this SAT-based technique is efficient in solving industrial problems. In fact, all state-of-the-art MaxSAT solvers that perform well in industrial instances are based on this technique. In particular, our solvers won the 2009 partial MaxSAT and the 2011 weighted partial MaxSAT industrial categories of the MaxSAT evaluation. We prove the correctness of all our algorithms. We also present a complete experimental study comparing the performance of our algorithms with latest MaxSAT solvers.

On the Relative Complexity of Resolution Refinements and Cutting Planes Proof Systems

An exponential lower bound for the size of tree-like cutting planes refutations of a certain family of conjunctive normal form (CNF) formulas with polynomial size resolution refutations is proved. This implies an exponential separation between the tree-like versions and the dag-like versions of resolution and cutting planes. In both cases only superpolynomial separations were known [A. Urquhart, Bull. Symbolic Logic, 1 (1995), pp. 425--467; J. Johannsen, Inform. Process. Lett., 67 (1998), pp. 37--41; P. Clote and A. Setzer, in Proof Complexity and Feasible Arithmetics, Amer. Math. Soc., Providence, RI, 1998, pp. 93--117]. In order to prove these separations, the lower bounds on the depth of monotone circuits of Raz and McKenzie in [ Combinatorica, 19 (1999), pp. 403--435] are extended to monotone real circuits. An exponential separation is also proved between tree-like resolution and several refinements of resolution: negative resolution and regular resolution. Actually, this last separation also provides a separation between tree-like resolution and ordered resolution, and thus the corresponding superpolynomial separation of [A. Urquhart, Bull. Symbolic Logic, 1 (1995), pp. 425--467] is extended. Finally, an exponential separation between ordered resolution and unrestricted resolution (also negative resolution) is proved. Only a superpolynomial separation between ordered and unrestricted resolution was previously known [A. Goerdt, Ann. Math. Artificial Intelligence, 6 (1992), pp. 169--184].

A study of proof search algorithms for resolution and polynomial calculus

The paper is concerned with the complexity of proofs and of searching for proofs in two propositional proof systems: Resolution and Polynomial Calculus (PC). For the former system we show that the recently proposed algorithm of E. Ben-Sasson and A. Wigderson (1999) for searching for proofs cannot give better than weakly exponential performance. This is a consequence of showing optimality of their general relationship, referred to as size-width trade-off. We moreover obtain the optimality of the size width trade-off for the widely used restrictions of resolution: regular, Davis-Putnam, negative, positive and linear. As for the second system, we show that the direct translation to polynomials of a CNF formula having short resolution proofs, cannot be refuted in PC with degree less than /spl Omega/ (log n). A consequence of this is that the simulation of resolution by PC of M. Clegg, J. Edmonds and R. Impagliazzo (1996) cannot be improved to better than quasipolynomial in the case where we start with small resolution proofs. We conjecture that the simulation of M. Clegg et al. is optimal.

Optimality of size-width tradeoffs for resolution

Abstract.This paper is concerned with the complexity of proofs and of searching for proofs in resolution. We show that the recently proposed algorithm of Ben-Sasson & Wigderson for searching for proofs in resolution cannot give better than weakly exponential performance. This is a consequence of our main result: we show the optimality of the general relationship called size-width tradeoffs in Ben-Sasson & Wigderson. Moreover we obtain the optimality of the size-width tradeoffs for the widely used restrictions of resolution: regular, Davis-Putnam, negative, positive.

Are there Hard Examples for Frege Systems

It is generally conjectured that there is an exponential separation between Frege and extended Frege systems. This paper reviews and introduces some candidates for families of combinatorial tautologies for which Frege proofs might need to be superpolynomially longer than extended Frege proofs. Surprisingly, we conclude that no particularly good or convincing examples are known. The examples of combinatorial tautologies that we consider seem to give at most a quasipolynomial speed-up of extended Frege proofs over Frege proofs, with the sole possible exception of tautologies based on a theorem of Frankl.

On the automatizability of resolution and related propositional proof systems

A propositional proof system is automatizable if there is an algorithm that, given a tautology, produces a proof in time polynomial in the size of its smallest proof. This notion can be weakened if we allow the algorithm to produce a proof in a stronger system within the same time bound. This new notion is called weak automatizability. Among other characterizations, we prove that a system is weakly automatizable exactly when a weak form of the satisfiability problem is solvable in polynomial time. After studying the robustness of the definition, we prove the equivalence between: (i) Resolution is weakly automatizable, (ii) Res(k) is weakly automatizable, and (iii) Res(k) has feasible interpolation, when k > 1. In order to prove this result, we show that Res(2) has polynomial-size proofs of the reflection principle of Resolution, which is a version of consistency. We also show that Res(k), for every k > 1, proves its consistency in polynomial size, while Resolution does not. In fact, we show that Resolution proofs of its own consistency require almost exponential size. This gives a better lower bound for the monotone interpolation of Res(2) and a separation from Resolution as a byproduct. Our techniques also give us a way to obtain a large class of examples that have small Resolution refutations but require relatively large width. This answers a question of Alekhnovich and Razborov related to whether Resolution is automatizable in quasipolynomial-time.

Lower bounds for the weak Pigeonhole principle and random formulas beyond resolution

We work with an extension of Resolution, called Res(2), that allows clauses with conjunctions of two literals. In this system there are rules to introduce and eliminate such conjunctions. We prove that the weak pigeonhole principle PHPncn and random unsatisfiable CNF formulas require exponential-size proofs in this system. This is the strongest system beyond Resolution for which such lower bounds are known. As a consequence to the result about the weak pigeonhole principle, Res(log) is exponentially more powerful than Res(2). Also we prove that Resolution cannot polynomially simulate Res(2) and that Res(2) does not have feasible monotone interpolation solving an open problem posed by Krajicek.

Lower bounds for cutting planes proofs with small coefficients

We consider small-weight Cutting Planes (CP* ) proofs; that is, Cutting Planes (CP) proofs with coefficients up to Poly(n). We use the well known lower bounds fc)r monotone complexity to prove an exponential lower bound for the length of CP* proofs, for a family of tautologies baaed on the clique function. Because Resolution is a special case of small-weight CP, our method also gives a new and simpler exponential lower bound fcn Resolution. We also prove the following two theorems : (1) Treelike CP’ proofs cannot polynomially simulate non-treelike CP* proofs. (2) Tree-like CP” proofs and Boundeddepth-Frege proofs cannot polynomially simulate each other. Our proofs also work for some generalizations of the C’P* proof system. In particular, they work for CP* with a deduction rule, and also for any proof system that allows any formula with small communication complexity, and any set of sound rules of inference.