167 research outputs found
On Tackling the Limits of Resolution in SAT Solving
The practical success of Boolean Satisfiability (SAT) solvers stems from the
CDCL (Conflict-Driven Clause Learning) approach to SAT solving. However, from a
propositional proof complexity perspective, CDCL is no more powerful than the
resolution proof system, for which many hard examples exist. This paper
proposes a new problem transformation, which enables reducing the decision
problem for formulas in conjunctive normal form (CNF) to the problem of solving
maximum satisfiability over Horn formulas. Given the new transformation, the
paper proves a polynomial bound on the number of MaxSAT resolution steps for
pigeonhole formulas. This result is in clear contrast with earlier results on
the length of proofs of MaxSAT resolution for pigeonhole formulas. The paper
also establishes the same polynomial bound in the case of modern core-guided
MaxSAT solvers. Experimental results, obtained on CNF formulas known to be hard
for CDCL SAT solvers, show that these can be efficiently solved with modern
MaxSAT solvers
Exploiting Resolution-based Representations for MaxSAT Solving
Most recent MaxSAT algorithms rely on a succession of calls to a SAT solver
in order to find an optimal solution. In particular, several algorithms take
advantage of the ability of SAT solvers to identify unsatisfiable subformulas.
Usually, these MaxSAT algorithms perform better when small unsatisfiable
subformulas are found early. However, this is not the case in many problem
instances, since the whole formula is given to the SAT solver in each call. In
this paper, we propose to partition the MaxSAT formula using a resolution-based
graph representation. Partitions are then iteratively joined by using a
proximity measure extracted from the graph representation of the formula. The
algorithm ends when only one partition remains and the optimal solution is
found. Experimental results show that this new approach further enhances a
state of the art MaxSAT solver to optimally solve a larger set of industrial
problem instances
Fuzzy Maximum Satisfiability
In this paper, we extend the Maximum Satisfiability (MaxSAT) problem to
{\L}ukasiewicz logic. The MaxSAT problem for a set of formulae {\Phi} is the
problem of finding an assignment to the variables in {\Phi} that satisfies the
maximum number of formulae. Three possible solutions (encodings) are proposed
to the new problem: (1) Disjunctive Linear Relations (DLRs), (2) Mixed Integer
Linear Programming (MILP) and (3) Weighted Constraint Satisfaction Problem
(WCSP). Like its Boolean counterpart, the extended fuzzy MaxSAT will have
numerous applications in optimization problems that involve vagueness.Comment: 10 page
Unifying Reasoning and Core-Guided Search for Maximum Satisfiability
A central algorithmic paradigm in maximum satisfiability solving geared towards real-world optimization problems is the core-guided approach. Furthermore, recent progress on preprocessing techniques is bringing in additional reasoning techniques to MaxSAT solving. Towards realizing their combined potential, understanding formal underpinnings of interleavings of preprocessing-style reasoning and core-guided algorithms is important. It turns out that earlier proposed notions for establishing correctness of core-guided algorithms and preprocessing, respectively, are not enough for capturing correctness of interleavings of the techniques. We provide an in-depth analysis of these and related MaxSAT instance transformations, and propose correction set reducibility as a notion that captures inprocessing MaxSAT solving within a state-transition style abstract MaxSAT solving framework. Furthermore, we establish a general theorem of correctness for applications of SAT-based preprocessing techniques in MaxSAT. The results pave way for generic techniques for arguing about the formal correctness of MaxSAT algorithms.Peer reviewe
On Optimization Modulo Theories, MaxSMT and Sorting Networks
Optimization Modulo Theories (OMT) is an extension of SMT which allows for
finding models that optimize given objectives. (Partial weighted) MaxSMT --or
equivalently OMT with Pseudo-Boolean objective functions, OMT+PB-- is a
very-relevant strict subcase of OMT. We classify existing approaches for MaxSMT
or OMT+PB in two groups: MaxSAT-based approaches exploit the efficiency of
state-of-the-art MAXSAT solvers, but they are specific-purpose and not always
applicable; OMT-based approaches are general-purpose, but they suffer from
intrinsic inefficiencies on MaxSMT/OMT+PB problems.
We identify a major source of such inefficiencies, and we address it by
enhancing OMT by means of bidirectional sorting networks. We implemented this
idea on top of the OptiMathSAT OMT solver. We run an extensive empirical
evaluation on a variety of problems, comparing MaxSAT-based and OMT-based
techniques, with and without sorting networks, implemented on top of
OptiMathSAT and {\nu}Z. The results support the effectiveness of this idea, and
provide interesting insights about the different approaches.Comment: 17 pages, submitted at Tacas 1
Core-Guided and Core-Boosted Search for CP
Peer reviewe
Clause Redundancy and Preprocessing in Maximum Satisfiability
The study of clause redundancy in Boolean satisfiability (SAT) has proven significant in various terms, from fundamental insights into preprocessing and inprocessing to the development of practical proof checkers and new types of strong proof systems. We study liftings of the recently-proposed notion of propagation redundancy-based on a semantic implication relationship between formulas-in the context of maximum satisfiability (MaxSAT), where of interest are reasoning techniques that preserve optimal cost (in contrast to preserving satisfiability in the realm of SAT). We establish that the strongest MaxSAT-lifting of propagation redundancy allows for changing in a controlled way the set of minimal correction sets in MaxSAT. This ability is key in succinctly expressing MaxSAT reasoning techniques and allows for obtaining correctness proofs in a uniform way for MaxSAT reasoning techniques very generally. Bridging theory to practice, we also provide a new MaxSAT preprocessor incorporating such extended techniques, and show through experiments its wide applicability in improving the performance of modern MaxSAT solvers.Peer reviewe
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