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On OBDDs for CNFs of bounded treewidth

by Igor Razgon - In KR , 2014
"... the read-once property of branching programs and ..."
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the read-once property of branching programs and
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Solving MaxSAT and #SAT on structured CNF formulas

by Sigve Hortemo Sæther, Jan Arne Telle, Martin Vatshelle - In C. Sinz & U. Egly (Eds.), SAT , 2014
"... 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 ..."
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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) algorithms for formulas F of m clauses and n variables and 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. 1
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Understanding model counting for β-acyclic CNF-formulas. CoRR, abs/1405.6043 . Retrieved from http://arxiv.org/ abs/1405.6043

by Johann Brault-baron, Florent Capelli, Stefan Mengel - Discrete Applied Mathematics , 2014
"... ar ..."
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Faster algorithms for vertex partitioning problems parameterized by clique-width

by Sang-il Oum, Sigve Hortemo Sæther, Martin Vatshelle , 2013
"... Many NP-hard problems, such as Dominating Set, are FPT parameterized by clique-width. For graphs of clique-width k given with a k-expression, Dominating Set can be solved in 4knO(1) time. However, no FPT algorithm is known for computing an optimal k-expression. For a graph of clique-width k, if we r ..."
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Many NP-hard problems, such as Dominating Set, are FPT parameterized by clique-width. For graphs of clique-width k given with a k-expression, Dominating Set can be solved in 4knO(1) time. However, no FPT algorithm is known for computing an optimal k-expression. For a graph of clique-width k, if we rely on known algorithms to compute a (23k − 1)-expression via rank-width and then solving Dominating Set using the (23k − 1)-expression, the above algorithm will only give a runtime of 42
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Between treewidth and clique-width

by Sigve Hortemo Sæther, Jan Arne Telle - in proceedings of WG 2014 , 2014
"... Many hard graph problems can be solved efficiently when restricted to graphs of bounded treewidth, and more generally to graphs of bounded clique-width. But there is a price to be paid for this generality, exemplified by the four problems MaxCut, Graph Coloring, Hamiltonian Cycle and Edge Dominating ..."
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Many hard graph problems can be solved efficiently when restricted to graphs of bounded treewidth, and more generally to graphs of bounded clique-width. But there is a price to be paid for this generality, exemplified by the four problems MaxCut, Graph Coloring, Hamiltonian Cycle and Edge Dominating Set that are all FPT parameterized by treewidth but none of which can be FPT parameterized by clique-width unless FPT = W[1], as shown by Fomin et al [7, 8]1. We therefore seek a struc-tural graph parameter that shares some of the generality of clique-width without paying this price. Based on splits, branch decompositions and the work of Vatshelle [18] on Maximum Matching-width, we consider the graph parameter sm-width which lies between treewidth and clique-width. Some graph classes of unbounded treewidth, like distance-hereditary graphs, have bounded sm-width. We show that MaxCut, Graph Coloring, Hamiltonian Cycle and Edge Dominating Set are all FPT parameterized by sm-width. 1
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Solving Hamiltonian Cycle by an EPT Algorithm for a Non-sparse Parameter

by Sigve Hortemo Sæther
"... Many hard graph problems, such as Hamiltonian Cycle, become FPT when parameterized by treewidth, a parameter that is bounded only on sparse graphs. When parameterized by the more general parameter clique-width, Hamiltonian Cycle becomes W[1]-hard, as shown by Fomin et al. [5]. Sæther and Telle addre ..."
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Many hard graph problems, such as Hamiltonian Cycle, become FPT when parameterized by treewidth, a parameter that is bounded only on sparse graphs. When parameterized by the more general parameter clique-width, Hamiltonian Cycle becomes W[1]-hard, as shown by Fomin et al. [5]. Sæther and Telle address this problem in their paper [13] by introducing a new parameter, split-matching-width, which lies between treewidth and clique-width in terms of generality. They show that even though graphs of restricted split-matching-width might be dense, solving problems such as Hamiltonian Cycle can be done in FPT time. Recently, it was shown that Hamiltonian Cycle parameterized by treewidth is in EPT [1, 6], meaning it can be solved in nO(1)2O(k)-time. In this paper, using tools from [6], we show that also parameterized by split-matching-width Hamiltonian Cycle is EPT. To the best of our knowl-edge, this is the first EPT algorithm for any ”globally constrained ” graph problem parameterized by a non-trivial and non-sparse structural parame-ter. To accomplish this, we also give an algorithm constructing a branch decomposition approximating the minimum split-matching-width to within a constant factor. Combined, these results show that the algorithms in [13] for Edge Dominating Set, Chromatic Number and Max Cut all can be improved. We also show that for Hamiltonian Cycle and Max Cut the resulting algorithms are asymptotically optimal under the Exponential Time Hypothesis. 1