On Partitioning for Maximum Satisfiability

Partitioning formulas is motivated by the expectation to identify easy to solve subformulas, even though at the cost of having more formulas to solve. In this paper we suggest to apply partitioning to Maximum Satisfiability (MaxSAT), the optimization version of the well-known Satisfiability (SAT) problem. The use of partitions can be naturally combined with unsatisfiability-based algorithms for MaxSAT that are built upon successive calls to a SAT solver, where each call identifies an unsatisfiable subformula. One of the drawbacks of these algorithms results from the SAT solver returning large unsatisfiable subformulas. However, when using partitions, the solver is more likely to identify smaller unsatisfiable subformulas. Experimental results show that the use of partitions in MaxSAT significantly improves the performance of unsatisfiability-based algorithms. 1 Maximum Satisfiability Maximum Satisfiability (MaxSAT) can be seen as an optimization version of Boolean Satisfiability (SAT) which consists in finding an assignment to the variables such that it minimizes (maximizes) the number of unsatisfied (satisfied) clauses. MaxSAT has several variants and can be generalized to the weighted partial MaxSAT problem. In this problem, some clauses are declared as hard, while the rest are declared as soft. The objective is to find an assignment to the variables such that all hard clauses are satisfied, while minimizing the total weight of unsatisfied soft clauses. For a more detailed introduction to MaxSAT we point the reader to the literature [5]. Unsatisfiability-based algorithms for weighted partial MaxSAT [6, 1, 2, 3] iteratively identify and relax unsatisfiable subformulas. In this paper we propose to improve these algorithms by using a new technique based on partitioning soft clauses. Instead of using the initial weighted partial MaxSAT formula to search for unsatisfiable subformulas, this paper proposes to start with a smaller formula that only contains a partition of the soft clauses. At each iteration, the formula is constrained by adding one more partition of soft clauses. This procedure is repeated until all partitions are added to the formula. The motivation for this technique is twofold. First, at each iteration we are solving formulas that are less constrained than the initial formula. Although the number of iterations may be larger than when not using partitions, each iteration is expected to require less time. As a result, more iterations may not imply using more computational time at the end. Second, by splitting soft clauses into partitions, the search is focused on a given subset of soft clauses. This can lead to finding smaller unsatisfiable subformulas that are less likely to be found if we consider the whole set of soft clauses. 1 INESC-ID/IST, TU Lisbon, Portugal, {ruben, vmm, ines}@sat.inescid.pt. This work was partially supported by FCT under research projects PTDC/EIACCO/102077/2008 and PTDC/EIA-CCO/110921/2009, and INESC-ID multiannual funding through the PIDDAC program funds. Algorithm 1: Unsatisfiability-based algorithm for weighted partial MaxSAT enhanced with partitioning of soft clauses Input: φ = φh ∪ φs Output: satisfiable assignment to φ or UNSAT 1 γ ← 〈γ1, . . . , γn〉 ← partitionSoft(φs) 2 φW ← φh 3 while true do 4 φW ← φW ∪ first(γ)