We present an exact algorithm that decides, for every fixed r ≥ 2 in time O(m) + 2 O(k 2 ) whether a given multiset of m clauses of size r admits a truth assignment that satisfies at least ((2 r − 1)m + k)/2 r clauses. Thus Max-rSat is fixed-parameter tractable when parameterized by the number of satisfied clauses above the tight lower bound (1 − 2 −r )m. This solves an open problem of Mahajan, Raman and Sikdar (J. Comput. System Sci., 75, 2009).Our algorithm is based on a polynomial-time data reduction procedure that reduces a problem instance to an equivalent algebraically represented problem with O(k 2 ) variables. This is done by representing the instance as an appropriate polynomial, and by applying a probabilistic argument combined with some simple tools from Harmonic analysis to show that if the polynomial cannot be reduced to one of size O(k 2 ), then there is a truth assignment satisfying the required number of clauses.We introduce a new notion of bikernelization from a parameterized problem to another one and apply it to prove that the above-mentioned parameterized Max-r-Sat admits a polynomial-size kernel.Combining another probabilistic argument with tools from graph matching theory and signed graphs, we show that if an instance of Max-2-Sat with m clauses has at least 3k variables after application of certain polynomial time reduction rules to it, then there is a truth assignment that satisfies at least (3m + k)/4 clauses.We also outline how the fixed-parameter tractability and polynomial-size kernel results on Max-r-Sat can be extended to more general families of Boolean Constraint Satisfaction Problems. * A preliminary version of this paper is
Abstract. We present algorithms for the propositional model counting problem #SAT. The algorithms are based on tree-decompositions of graphs associated with the given CNF formula, in particular primal, dual, and incidence graphs. We describe the algorithms in a coherent fashion that admits a direct comparison of their algorithmic advantages. We analyze and discuss several aspects of the algorithms including worst-case time and space requirements and simplicity of implementation. The algorithms are described in sufficient detail for making an implementation reasonably easy.
We present an exact algorithm that decides, for every fixed Our algorithm is based on a polynomial-time data reduction procedure that reduces a problem instance to an equivalent algebraically represented problem with O(k 2 ) variables. This is done by representing the instance as an appropriate polynomial, and by applying a probabilistic argument combined with some simple tools from Harmonic analysis to show that if the polynomial cannot be reduced to one of size O(k 2 ), then there is a truth assignment satisfying the required number of clauses.Combining another probabilistic argument with tools from graph matching theory and signed graphs, we show that if an instance of Max-2-Sat with m clauses has at least 3k variables after application of certain polynomial time reduction rules to it, then there is a truth assignment that satisfies at least (3m + k)/4 clauses.We also outline how the fixed-parameter tractability result on Max-r-Sat can be extended to a family of Boolean Constraint Satisfaction Problems.
This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are encouraged to visit: In this paper, we study the complexity of several coloring problems on graphs, parameterized by the treewidth of the graph. 1. The List Coloring problem takes as input a graph G, together with an assignment to each vertex v of a set of colors C v. The problem is to determine whether it is possible to choose a color for vertex v from the set of permitted colors C v , for each vertex, so that the obtained coloring of G is proper. We show that this problem is W [1]-hard, parameterized by the treewidth of G. The closely related Precoloring Extension problem is also shown to be W [1]-hard, parameterized by treewidth. 2. An equitable coloring of a graph G is a proper coloring of the vertices where the numbers of vertices having any two distinct colors differs by at most one. We show that the problem is hard for W [1], parameterized by the treewidth plus the number of colors. We also show that a list-based variation, List Equitable Coloring is W [1]-hard for forests, parameter-ized by the number of colors on the lists. 3. The list chromatic number χ l (G) of a graph G is defined to be the smallest positive integer r, such that for every assignment to the vertices v of G, of a list L v of colors, where each list has length at least r, there is a choice of one color from each vertex list L v yielding a proper coloring of G. We show that the problem of determining whether χ l (G) ≤ r, the List Chromatic Number problem, is solvable in linear time on graphs of constant treewidth.
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