Maximum Satisfiability (MaxSAT) and its weighted and partial variants are well-known optimization formulations of Boolean Satisfiability (SAT). MaxSAT consists of finding an assignment that satisfies the (possibly empty) set of hard clauses, while minimizing the sum of weights of the falsified soft clauses. Recent years have witnessed the development of complete algorithms for MaxSAT motivated by a number of practical applications. The most effective approaches in such practical settings are based on iteratively calling a SAT solver and computing unsatisfiable cores to guide the search. Such approaches use computed unsatisfiable cores from unsatisfiable (UNSAT) outcomes to relax the soft clauses occurring in the computed cores. Surprisingly, only recently has an approach been proposed that exploits models from satisfiable (SAT) outcomes [1], [2] rather than unsatisfiable cores from UNSAT outcomes. This paper proposes two novel MaxSAT algorithms which exploit SAT outcomes to relax soft clauses taking into account the computed models. The new algorithms are shown to outperform classical MaxSAT algorithms and to be fairly competitive with recent core-guided MaxSAT algorithms. Finally, a well-known core-guided MaxSAT algorithm is extended to additionally exploit computed models in an attempt to integrate both approaches.

Minimum Satisfiability (MinSAT) denotes one of the optimization versions of the Boolean Satisfiability (SAT) problem. In some settings MinSAT is preferred to using Maximum Satis-fiability (MaxSAT). Several encodings and dedicated branch and bound algorithms for MinSAT have been recently proposed, and evaluated on small challenging randomly generated instances. Motivated by the observation that current best performing MaxSAT algorithms for structured and industrial instances are based on computing unsatisfiable cores with a SAT solver, this paper proposes novel approaches for MinSAT, that also target these instances. First, the paper proposes an algorithm based on iteratively calling a SAT solver which uses the computed models to relax clauses. Second, the paper proposes group-based MinSAT solving, which is essentially a novel reduction of the MinSAT problem into the Group MaxSAT problem. For a given MinSAT instance, the resulting Group MaxSAT formula is then translated into a standard MaxSAT formula which specifically targets unsatisfiability-based MaxSAT algorithms. Experimental results indicate that, similarly to MaxSAT, the proposed approaches outperform branch and bound algorithms on problem instances obtained from practical applications.

Several MaxSAT algorithms based on iterative SAT solving have been proposed in recent years. These algorithms are in general the most efficient for real-world applications. Existing data indicates that, among MaxSAT algorithms based on iterative SAT solving, the most efficient ones are core-guided, i.e. algorithms which guide the search by iteratively computing unsatisfiable subformulas (or cores). For weighted MaxSAT, core-guided algorithms exhibit a number of important drawbacks, including a possibly exponential number of iterations and the use of a large number of auxiliary variables. This paper develops two new algorithms for (weighted) MaxSAT that address these two drawbacks. The first MaxSAT algorithm implements core-guided iterative SAT solving with binary search. The second algorithm extends the first one by exploiting disjoint cores. The empirical evaluation shows that core-guided binary search is competitive with current MaxSAT solvers.

This paper proposes the integration of the resolution rule for Max-SAT with unsatisfiability-based Max-SAT solvers. First, we show that the resolution rule for Max-SAT can be safely applied as dictated by the resolution proof associated with an unsatisfiable core when such proof is read-once, that is, each clause is used at most once in the resolution process. Second, we study how this property can be integrated in an unsatisfiability-based solver. In particular, the resolution rule for Max-SAT is applied to read-once proofs or to read-once subparts of a general proof. Finally, we perform an empirical investigation on structured instances from recent Max-SAT evaluations. Preliminary results show that the use of read-once resolution substantially improves the performance of the solver.

In this paper we introduce MiniMaxSat, a new Max-SAT solver that is built on top of MiniSat+. It incorporates the best current SAT and Max-SAT techniques. It can handle hard clauses(clauses of mandatory satisfaction as in SAT), soft clauses (clauses whose falsification is penalized by a cost as in Max-SAT) as well as pseudo-boolean objective functions and constraints. Its main features are: learning and backjumping on hard clauses; resolution-based and substraction-based lower bounding; and lazy propagation with the two-watched literal scheme. Our empirical evaluation comparing a wide set of solving alternatives on a broad set of optimization benchmarks indicates that the performance of MiniMaxSat is usually close to the best specialized alternative and, in some cases, even better.