MaxSAT Solvers Research Articles

Dynamic SAT with Decision Change Costs: Formalization and Solutions

We address a dynamic decision problem in which decision makers must pay some costs when they change their decisions along the way. We formalize this problem as Dynamic SAT (DynSAT) with decision change costs, whose goal is to find a sequence of models that minimize the aggregation of the costs for changing variables. We provide two solutions to solve a specific case of this problem. The first uses a Weighted Partial MaxSAT solver after we encode the entire problem as a Weighted Partial MaxSAT problem. The second solution, which we believe is novel, uses the Lagrangian decomposition technique that divides the entire problem into sub-problems, each of which can be separately solved by an exact Weighted Partial MaxSAT solver, and produces both lower and upper bounds on the optimal in an anytime manner. To compare the performance of these solvers, we experimented on the random problem and the target tracking problem. The experimental results show that a solver based on Lagrangian decomposition performs better for the random problem and competitively for the target tracking problem.

Max-SAT formalisms with hard and soft constraints

The study of Max-SAT formalisms and the development of fast Max-SAT solvers have become very active research topics in the last few years. This PhD dissertation focus on the design, implementation ...

Resolution-based lower bounds in MaxSAT

The lower bound (LB) implemented in branch and bound MaxSAT solvers is decisive for obtaining a competitive solver. In modern solvers like MaxSatz and MiniMaxSat, the LB relies on the cooperation of the underestimation and inference components. Actually, the underestimation component of some solvers guides the application of the inference component when a conflict is reached and certain structures are found. In this paper we analyze algorithmic and logical aspects of the underestimation components that have been implemented in MaxSatz during its evolution. From an algorithmic point of view, we define novel strategies for selecting unit clauses in UP (the underestimation of LB in UP is the number of independent inconsistent subformulas detected using unit propagation), the extension of UP with failed literal detection, and a clever heuristic for guiding the application of MaxSAT resolution when UP augmented with failed literal detection is applied in the presence of cycles structures. From a logical point of view, we prove that the inconsistent subformulas detected by UP are minimally unsatisfiable, but this property does not hold if failed literal detection is added. In the presence of cycle structures in conflicts detected by UP augmented with failed literal detection, we prove that the application of MaxSAT resolution produces smaller inconsistent subformulas and, besides, generates additional clauses that may be used to improve the LB. The conducted empirical investigation indicates that the LB techniques described in this paper lead to better quality LBs.

A New Algorithm for Weighted Partial MaxSAT

We present and implement a Weighted Partial MaxSAT solver based on successive calls to a SAT solver. We prove the correctness of our algorithm and compare our solver with other Weighted Partial MaxSAT solvers.

Terse Integer Linear Programs for Boolean Optimization

We present a new polyhedral approach to nonlinear Boolean optimization. Compared to other methods, it produces much smaller integer programming models, making it more efficient from a practical point of view. We mainly obtain this by two different ideas: first, we do not require the objective function to be in any normal form. The transformation into a normal form usually leads to the introduction of many additional variables or constraints. Second, we reduce the problem to the degree-two case in a very efficient way, by slightly extending the dimension of the original variable space. The resulting model turns out to be closely related to the maximum cut problem; we show that the corresponding polytope is a face of a suitable cut polytope in most cases. In particular, our separation problem reduces to the one for the maximum cut problem. In practice, the approach appears to be very competitive for unconstrained Boolean optimization problems. First experimental results, which have been obtained for some particularly hard instances of the Max-SAT Evaluation 2007, show that our very general implementation can outperform even special-purpose Max-SAT solvers. The software is accessible online under “we.logoptimize.it”.

Guest Editors Conclusion

This special issue comprises six papers devoted to SAT solving and three papers devoted to MAXSAT solving. With two exception, the papers contain detailed information about some of the best solvers from the 5th SAT solver competition and from the 2nd MAX-SAT solver competition, which took place as side event of the 10th International Conference on Theory and Applications of Satisfiability Testing SAT 2007, in Lisboa/ Portugal. The paper by Heras, et al. deals with the contributed instances for the MAX-SAT evaluations and the paper by Argelich, et al. reports on the First and Second MAX-SAT evaluations in 2006 and 2007. The first paper PicoSAT Essentials presents three nice concepts: optimized compact data structures for watching literals, a new restart strategy, and an efficient proof encoding. Experiments demonstrate, that this low level optimization saves memory and speeds up the SAT solver considerably. The second paper Parallel SAT Solving using Bit-level Operations describes a technique for speeding up local search SAT solvers by using multi-bit Boolean operations, leveraging that k-bit CPUs can do k-bit Boolean operations in one time step. Modifying the state-of-the-art local search SAT solver Unit Walk according to this technic yields a much faster implementation than the original one. The third paper Whose side are you on? Finding solutions in a biased search-tree uses so called direction heuristics for selecting good assignments to the decision variables in order to guide the search into those parts of the search tree that are more likely to contain solutions. The SAT solver march ks, based on this idea was the winner of the crafted category during the SAT 2007 competition.

Solving Weighted Max-SAT Problems in a Reduced Search Space: A Performance Analysis1

We analyze, in this work, the performance of a recently introduced weighted Max-SAT solver, Clone, in the Max-SAT evaluation 2007. Clone utilizes a novel bound computation based on formula compilation that allows it to search in a reduced search space. We study how additional techniques from the SAT and Max-SAT literature affect the performance of Clone on problems from the evaluation. We then perform further investigations on factors that may affect the performance of leading Max-SAT solvers. We empirically identify two properties of weighted Max-SAT problems that can be used to adjust the difficulty level of the problems with respect to the considered solvers.

MiniMaxSAT: An Efficient Weighted Max-SAT 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.

New Inference Rules for Max-SAT

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. With the aim of finding 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 provide empirical evidence that MaxSatz is very competitive, at least, on random Max-2SAT, random Max-3SAT, Max-Cut, and Graph 3-coloring instances, as well as on the benchmarks from the Max-SAT Evaluation 2006.

ARMS: an automatic knowledge engineering tool for learning action models for AI planning

AbstractWe present an action model learning system known as ARMS (Action-Relation Modelling System) for automatically discovering action models from a set of successfully observed plans. Current artificial intelligence (AI) planners show impressive performance in many real world and artificial domains, but they all require the definition of an action model. ARMS is aimed at automatically learning action models from observed example plans, where each example plan is a sequence of action traces. These action models can then be used by the human editors to refine. The expectation is that this system will lessen the burden of the human editors in designing action models from scratch. In this paper, we describe the ARMS in detail. To learn action models, ARMS gathers knowledge on the statistical distribution of frequent sets of actions in the example plans. It then builds a weighted propositional satisfiability (weighted SAT) problem and solves it using a weighted MAXSAT solver. Furthermore, we show empirical evidence that ARMS can indeed learn a good approximation of the finally action models effectively.

A logical approach to efficient Max-SAT solving

Weighted Max-SAT is the optimization version of SAT and many important problems can be naturally encoded as such. Solving weighted Max-SAT is an important problem from both a theoretical and a practical point of view. In recent years, there has been considerable interest in finding efficient solving techniques. Most of this work focuses on the computation of good quality lower bounds to be used within a branch and bound DPLL-like algorithm. Most often, these lower bounds are described in a procedural way. Because of that, it is difficult to realize the logic that is behind. In this paper we introduce an original framework for Max-SAT that stresses the parallelism with classical SAT. Then, we extend the two basic SAT solving techniques: search and inference. We show that many algorithmic tricks used in state-of-the-art Max-SAT solvers are easily expressible in logical terms in a unified manner, using our framework. We also introduce an original search algorithm that performs a restricted amount of weighted resolution at each visited node. We empirically compare our algorithm with a variety of solving alternatives on several benchmarks. Our experiments, which constitute to the best of our knowledge the most comprehensive Max-SAT evaluation ever reported, demonstrate the practical usability of our approach.

Learning action models from plan examples using weighted MAX-SAT

AI planning requires the definition of action models using a formal action and plan description language, such as the standard Planning Domain Definition Language (PDDL), as input. However, building action models from scratch is a difficult and time-consuming task, even for experts. In this paper, we develop an algorithm called ARMS (action-relation modelling system) for automatically discovering action models from a set of successful observed plans. Unlike the previous work in action-model learning, we do not assume complete knowledge of states in the middle of observed plans. In fact, our approach works when no or partial intermediate states are given. These example plans are obtained by an observation agent who does not know the logical encoding of the actions and the full state information between the actions. In a real world application, the cost is prohibitively high in labelling the training examples by manually annotating every state in a plan example from snapshots of an environment. To learn action models, ARMS gathers knowledge on the statistical distribution of frequent sets of actions in the example plans. It then builds a weighted propositional satisfiability (weighted MAX-SAT) problem and solves it using a MAX-SAT solver. We lay the theoretical foundations of the learning problem and evaluate the effectiveness of ARMS empirically.

Exact Max-SAT solvers for over-constrained problems

We present a new generic problem solving approach for over-constrained problems based on Max-SAT. We first define a Boolean clausal form formalism, called soft CNF formulas, that deals with blocks of clauses instead of individual clauses, and that allows one to declare each block either as hard (i.e., must be satisfied by any solution) or soft (i.e., can be violated by some solution). We then present two Max-SAT solvers that find a truth assignment that satisfies all the hard blocks of clauses and the maximum number of soft blocks of clauses. Our solvers are branch and bound algorithms equipped with original lazy data structures, powerful inference techniques, good quality lower bounds, and original variable selection heuristics. Finally, we report an experimental investigation on a representative sample of instances (random 2-SAT, Max-CSP, graph coloring, pigeon hole and quasigroup completion) which provides experimental evidence that our approach is very competitive compared with the state-of-the-art approaches developed in the CSP and SAT communities.

MaxSolver: An efficient exact algorithm for (weighted) maximum satisfiability

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 binary-clause first rule and a dynamic-weighting 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.