31 research outputs found

    Solving MaxSAT and #SAT on structured CNF formulas

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    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 precisely satisfiable if there is some complete assignment satisfying these clauses only. Let the ps-value of the formula be the number of precisely 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 O(m2(m+n)s)O(m^2(m + n)s) algorithms for formulas FF of mm clauses and nn variables and size ss, if FF has a linear ordering of the variables and clauses such that for any variable xx occurring in clause CC, if xx appears before CC then any variable between them also occurs in CC, and if CC appears before xx then xx occurs also in any clause between them. Note that the class of incidence graphs of such formulas do not have bounded clique-width

    On the Kernel and Related Problems in Interval Digraphs

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    Given a digraph GG, a set XV(G)X\subseteq V(G) is said to be absorbing set (resp. dominating set) if every vertex in the graph is either in XX or is an in-neighbour (resp. out-neighbour) of a vertex in XX. A set SV(G)S\subseteq V(G) is said to be an independent set if no two vertices in SS are adjacent in GG. A kernel (resp. solution) of GG is an independent and absorbing (resp. dominating) set in GG. We explore the algorithmic complexity of these problems in the well known class of interval digraphs. A digraph GG is an interval digraph if a pair of intervals (Su,Tu)(S_u,T_u) can be assigned to each vertex uu of GG such that (u,v)E(G)(u,v)\in E(G) if and only if SuTvS_u\cap T_v\neq\emptyset. Many different subclasses of interval digraphs have been defined and studied in the literature by restricting the kinds of pairs of intervals that can be assigned to the vertices. We observe that several of these classes, like interval catch digraphs, interval nest digraphs, adjusted interval digraphs and chronological interval digraphs, are subclasses of the more general class of reflexive interval digraphs -- which arise when we require that the two intervals assigned to a vertex have to intersect. We show that all the problems mentioned above are efficiently solvable, in most of the cases even linear-time solvable, in the class of reflexive interval digraphs, but are APX-hard on even the very restricted class of interval digraphs called point-point digraphs, where the two intervals assigned to each vertex are required to be degenerate, i.e. they consist of a single point each. The results we obtain improve and generalize several existing algorithms and structural results for subclasses of reflexive interval digraphs.Comment: 26 pages, 3 figure

    On Satisfiability Problems with a Linear Structure

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