# Solving MaxSAT and #SAT on Structured CNF Formulas

• Sigve Hortemo Sæther
• Jan Arne Telle
• Martin Vatshelle
Conference paper

DOI: 10.1007/978-3-319-09284-3_3

Part of the Lecture Notes in Computer Science book series (LNCS, volume 8561)
Cite this paper as:
Sæther S.H., Telle J.A., Vatshelle M. (2014) Solving MaxSAT and #SAT on Structured CNF Formulas. In: Sinz C., Egly U. (eds) Theory and Applications of Satisfiability Testing – SAT 2014. SAT 2014. Lecture Notes in Computer Science, vol 8561. Springer, Cham

## Abstract

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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© 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