Solving MaxSAT and #SAT on Structured CNF Formulas

  • Sigve Hortemo Sæther
  • Jan Arne Telle
  • Martin Vatshelle
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8561)


In this paper we propose a structural parameter of CNF formulas and use it to identify instances of weighted MaxSAT and #SAT that can be solved in polynomial time. Given a CNF formula we say that a set of clauses is projection satisfiable if there is some complete assignment satisfying these clauses only. Let the ps-value of the formula be the number of projection satisfiable sets of clauses. Applying the notion of branch decompositions to CNF formulas and using ps-value as cut function, we define the ps-width of a formula. For a formula given with a decomposition of polynomial ps-width we show dynamic programming algorithms solving weighted MaxSAT and #SAT in polynomial time. Combining with results of ’Belmonte and Vatshelle, Graph classes with structured neighborhoods and algorithmic applications, Theor. Comput. Sci. 511: 54-65 (2013)’ we get polynomial-time algorithms solving weighted MaxSAT and #SAT for some classes of structured CNF formulas. For example, we get \({\mathcal O}(m^2(m + n)s)\) algorithms for formulas F of m clauses and n variables and total size s, if F has a linear ordering of the variables and clauses such that for any variable x occurring in clause C, if x appears before C then any variable between them also occurs in C, and if C appears before x then x occurs also in any clause between them. Note that the class of incidence graphs of such formulas do not have bounded clique-width.


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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Sigve Hortemo Sæther
    • 1
  • Jan Arne Telle
    • 1
  • Martin Vatshelle
    • 1
  1. 1.Department of InformaticsUniversity of BergenNorway

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