Artificial Intelligence, to appear Maximum Boolean satisfiability (max-SAT) is the optimization counterpart of Boolean satisfiability (SAT), in which a variable assignment is sought to satisfy the maximum number of clauses in a Boolean formula. A branch and bound algorithm based on the Davis-Putnam-Logemann-Loveland procedure (DPLL) is one of the most competitive exact algorithms for solving max-SAT. In this paper, we propose and investigate a number of strategies for max-SAT. The first strategy is a set of unit propagation or unit resolution rules for max-SAT. We summarize three existing unit propagation rules and propose a new one based on a nonlinear programming formulation of max-SAT. The second strategy is an effective lower bound based on linear programming (LP). We show that the LP lower bound can be made effective as the number of clauses increases. The third strategy consists of a a binary-clause first rule and a dynamicweighting variable ordering rule, which are motivated by a thorough analysis of two existing well-known variable orderings. Based on the analysis of these strategies, we develop an exact solver for both max-SAT and weighted max-SAT. Our experimental results on random problem instances and many instances from the max-SAT libraries show that our new solver outperforms most of the existing exact max-SAT solvers, with orders of magnitude of improvement in many cases.

Abstract. We present two new branch and bound weighted Max-SAT solvers (Lazy and Lazy ⋆ ) which incorporate original data structures and inference rules, and a lower bound of better quality. 1

Abstract. Exact Max-SAT solvers, compared with SAT solvers, apply little inference at each node of the proof tree. Commonly used SAT inference rules like unit propagation produce a simplified formula that preserves satisfiability but, unfortunately, solving the Max-SAT problem for the simplified formula is not equivalent to solving it for the original formula. In this paper, we define a number of original inference rules that, besides being applied efficiently, transform Max-SAT instances into equivalent Max-SAT instances which are easier to solve. The soundness of the rules, that can be seen as refinements of unit resolution adapted to Max-SAT, are proved in a novel and simple way via an integer programming transformation. Aiming to find out how powerful the inference rules are in practice, we have developed a new Max-SAT solver, called MaxSatz, which incorporates those rules, and performed an experimental investigation. The results obtained provide empirical evidence that MaxSatz is very competitive and greatly outperforms the best stateof-the-art Max-SAT solvers on random Max-2SAT, random Max-3SAT, Max-Cut, and Graph 3-coloring instances, as well as benchmarks submitted to the Max-SAT Evaluation 2006. 1

Abstract. Max-SAT is the problem of finding an assignment minimizing the number of unsatisfied clauses of a given CNF formula. We propose a resolution-like calculus for Max-SAT and prove its soundness and completeness. We also prove the completeness of some refinements of this calculus. From the completeness proof we derive an exact algorithm for Max-SAT and a time upper bound. 1

Given a Boolean satisfiability (Sat) problem whose variables have non-negative weights, the minimum-cost satisfiability (MinCostSat) problem finds a satisfying truth assignment that minimizes a weighted sum of the truth values of the variables. Many NP-optimization problems are either special cases of MinCostSat or can be transformed into MinCostSat efficiently. However, in the past, these problems have been largely considered in isolation. In this dissertation, we (1) classify existing Min-CostSat problems, (2) study factors affecting the performance of MinCostSat solvers, (3) propose algorithms for MinCostSat problems, and (4) implement and validate the performance of state-of-the-art solvers for special cases of MinCostSat, including set and binate covering, Max-Sat, and group-partial Max-Sat. We categorize MinCostSat problems as either native or non-native. Non-native problems can only be transformed into MinCostSat by adding slack variables. These problems include the Max-Sat, partial Max-Sat, and group-partial Max-Sat problems which have applications ranging from course assignment to FPGA detailed routing. Native problems are various sub-cases of MinCostSat. We further divide these into two

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Abstract. We present a new branch and bound algorithm for Max-SAT which incorporates original lazy data structures, a new variable selection heuristics and a lower bound of better quality. We provide experimental evidence that our solver outperforms some of the best performing Max-SAT solvers on a wide range of instances. Keywords: Max-SAT, branch and bound, lower bound, heuristics, data structure. 1

Abstract. We give an exact deterministic algorithm for MAX-SAT. On input CNF formulas with constant clause density (the ratio of the number of clauses to the number of variables is a constant), this algorithm runs in O(c n) time where c < 2 and n is the number of variables. Worst-case upper bounds for MAX-SAT less than O(2 n) were previously known only for k-CNF formulas and for CNF formulas with small clause density. 1

We define the MaxSAT problem for many-valued CNF formulas, called many-valued MaxSAT, and establish its complexity class. We then describe a basic branch and bound algorithm for solving many-valued MaxSAT, and an exact many-valued MaxSAT solver we have implemented. Finally, we report the experimental investigation we have performed to compare our solver with Boolean MaxSAT solvers on graph coloring instances. The results obtained indicate that many-valued CNF formulas can become a competitive formalism for representing and solving combinatorial optimization problems. 1

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