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Resolution-based lower bounds in MaxSAT

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Abstract

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.

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Authors and Affiliations

  1. Hunan Normal University, Changsha, China

    Chu Min Li

  2. MIS, Université de Picardie Jules Verne, 33 Rue St. Leu, 80039, Amiens Cedex 01, France

    Chu Min Li & Nouredine Ould Mohamedou

  3. Artificial Intelligence Research Institute (IIIA, CSIC), Campus UAB, 08193, Bellaterra, Spain

    Felip Manyà

  4. Computer Science Department, Universitat de Lleida, Jaume II, 69, 25001, Lleida, Spain

    Jordi Planes

Authors
  1. Chu Min Li
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  2. Felip Manyà
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  3. Nouredine Ould Mohamedou
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  4. Jordi Planes
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Corresponding author

Correspondence to Felip Manyà.

Additional information

Research partially supported by the Generalitat de Catalunya under grant 2009-SGR-1434, the Ministerio de Ciencia e Innovación research projects CONSOLIDER CSD2007-0022, INGENIO 2010, Acción Integrada HA2008-0017, TIN2007-68005-C04-04 and TIN2009-14704-C03-01, and the Ministerio de educación program Programa Nacional de Movilidad de Recursos Humanos.

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Li, C.M., Manyà, F., Mohamedou, N.O. et al. Resolution-based lower bounds in MaxSAT. Constraints 15, 456–484 (2010). https://doi.org/10.1007/s10601-010-9097-9

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