Combining approaches for solving Satisfiability problems with Preferences and their Evaluation Emanuele Di Rosa 1 , Enrico Giunchiglia 1 , and Barry O’Sullivan 2 1 DIST, Universit` a di Genova, Viale Causa, 13 – 16145 Genova, Italy {emanuele,enrico}@dist.unige.it 2 Cork Constraint Computation Centre Department of Computer Science, University College Cork, Ireland {b.osullivan@4c.ucc.ie} Abstract. The ability to effectively reason in the presence of qualitative prefer- ences on literals or formulas is a central issue in Artificial Intelligence. In the last few years, two procedures have been presented in order to reason with proposi- tional satisfiability (SAT) problems in the presence of additional, partially ordered qualitative preferences on literals or formulas: The first requires a modification of the branching heuristic of the SAT solver in order to guarantee that the first solu- tion is optimal, while the second computes a sequence of solutions, each guaran- teed to be better than the previous one. The two approaches have been compared on specific classes of instances —each having an empty partial order— showing that the second has superior performance. In this paper we show that these two approaches can be combined yielding a new effective procedure. In particular, we modify the branching heuristic —as in the first approach— by possibly changing the polarity of the returned literal, and then we continue the search —as in the second approach— looking for better solutions. We extended the experimental analysis conducted in previous papers by considering a wide variety of problems, having both an empty and a non-empty partial order: The results show that the new procedure performs better than the two previous approaches on average, and especially on the “hard” problems. 1 Introduction The ability to effectively reason in the presence of qualitative preferences on literals or formulas is a central issue in Artificial Intelligence. For example, in [2] qualitative pref- erences are represented as CP-nets and various examples of applications are discussed, while in [1, 7] qualitative preferences are used to impose additional “soft” constraints to be possibly satisfied while planning. In two previous papers [6, 10], it has been shown how it is possible to extend current state-of-the-art procedures for propositional satis- fiability (SAT) in order to compute an optimal model w.r.t. a given set of qualitative preferences, expressed as a partially ordered set 〈S, ≺〉 of literals or formulas: Intu- itively, S represents the preferences that we would like to have satisfied, ≺ models their relative importance, and a model µ of a formula ϕ is optimal if it is a minimal element of the partial order on the models of ϕ induced by 〈S, ≺〉. Given a formula ϕ and a (qualitative) preference 〈S, ≺〉, each of the two procedures presented in [6, 10] has its own pros and cons. The first, called OPTSAT- HS, requires