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Algorithms for propositional model counting.
 In Proc. of the 14th International Conference on Logic for Programming, Artificial Intelligence, and Reasoning (LPAR’07),
, 2007
"... Abstract We present algorithms for the propositional model counting problem #SAT. The algorithms utilize tree decompositions of certain graphs associated with the given CNF formula; in particular we consider primal, dual, and incidence graphs. We describe the algorithms coherently for a direct comp ..."
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Abstract We present algorithms for the propositional model counting problem #SAT. The algorithms utilize tree decompositions of certain graphs associated with the given CNF formula; in particular we consider primal, dual, and incidence graphs. We describe the algorithms coherently for a direct comparison and with sufficient detail for making an actual implementation reasonably easy. We discuss several aspects of the algorithms including worstcase time and space requirements.
Model Counting
, 2008
"... Propositional model counting or #SAT is the problem of computing the number of models for a given propositional formula, i.e., the number of distinct truth assignments to variables for which the formula evaluates to true. For a propositional formula F, we will use #F to denote the model count of F. ..."
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Cited by 24 (0 self)
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Propositional model counting or #SAT is the problem of computing the number of models for a given propositional formula, i.e., the number of distinct truth assignments to variables for which the formula evaluates to true. For a propositional formula F, we will use #F to denote the model count of F. This problem is also referred to as the solution counting problem for SAT. It generalizes SAT and is the canonical #Pcomplete problem. There has been significant theoretical work trying to characterize the worstcase complexity of counting problems, with some surprising results such as model counting being hard even for some polynomialtime solvable problems like 2SAT. The model counting problem presents fascinating challenges for practitioners and poses several new research questions. Efficient algorithms for this problem will have a significant impact on many application areas that are inherently beyond SAT (‘beyond ’ under standard complexity theoretic assumptions), such as boundedlength adversarial and contingency planning, and probabilistic reasoning. For example, various probabilistic inference problems, such as Bayesian net reasoning, can be effectively translated into model counting problems [cf.
Solving #SAT Using Vertex Covers
, 2006
"... We propose an exact algorithm for counting the models of propositional formulas in conjunctive normal form (CNF). Our algorithm is based on the detection of strong backdoor sets of bounded size; each instantiation of the variables of a strong backdoor set puts the given formula into a class of form ..."
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Cited by 21 (10 self)
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We propose an exact algorithm for counting the models of propositional formulas in conjunctive normal form (CNF). Our algorithm is based on the detection of strong backdoor sets of bounded size; each instantiation of the variables of a strong backdoor set puts the given formula into a class of formulas for which models can be counted in polynomial time. For the backdoor set detection we utilize an efficient vertex cover algorithm applied to a certain “obstruction graph ” that we associate with the given formula. This approach gives rise to a new hardness index for formulas, the clusteringwidth. Our algorithm runs in uniform polynomial time on formulas with bounded clusteringwidth. It is known that the number of models of formulas with bounded cliquewidth, bounded treewidth, or bounded branchwidth can be computed in polynomial time; these graph parameters are applied to formulas via certain (hyper)graphs associated with formulas. We show that clusteringwidth and the other parameters mentioned are incomparable: there are formulas with bounded clusteringwidth and arbitrarily large cliquewidth, treewidth, and branchwidth. Conversely, there are formulas with arbitrarily large clusteringwidth and bounded cliquewidth, treewidth, and branchwidth.
Bridging the gap between intensional and extensional query evaluation in probabilistic databases. EDBT
, 2010
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Backdoors to satisfaction
 The Multivariate Algorithmic Revolution and Beyond  Essays Dedicated to Michael R. Fellows on the Occasion of His 60th Birthday, volume 7370 of Lecture
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Computing graph polynomials on graphs of bounded cliquewidth
 GraphTheoretic Concepts in Computer Science, 32nd International Workshop, WG 2006
"... Abstract. We discuss the complexity of computing various graph polynomials of graphs of fixed cliquewidth. We show that the chromatic polynomial, the matching polynomial and the twovariable interlace polynomial of a graph G of cliquewidth at most k with n vertices can be computed in time O(n f(k) ..."
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Cited by 13 (5 self)
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Abstract. We discuss the complexity of computing various graph polynomials of graphs of fixed cliquewidth. We show that the chromatic polynomial, the matching polynomial and the twovariable interlace polynomial of a graph G of cliquewidth at most k with n vertices can be computed in time O(n f(k)), where f(k) ≤ 3 for the inerlace polynomial, f(k) ≤ 2k + 1 for the matching polynomial and f(k) ≤ 3 · 2 k+2 for the chromatic polynomial. 1
Satisfiability of acyclic and almost acyclic CNF formulas
 IN: FSTTCS
, 2010
"... We study the propositional satisfiability problem (SAT) on classes of CNF formulas (formulas in Conjunctive Normal Form) that obey certain structural restrictions in terms of their hypergraph structure, by associating to a CNF formula the hypergraph obtained by ignoring negations and considering cla ..."
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Cited by 10 (4 self)
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We study the propositional satisfiability problem (SAT) on classes of CNF formulas (formulas in Conjunctive Normal Form) that obey certain structural restrictions in terms of their hypergraph structure, by associating to a CNF formula the hypergraph obtained by ignoring negations and considering clauses as hyperedges on variables. We show that satisfiability of CNF formulas with socalled “βacyclic hypergraphs ” can be decided in polynomial time. We also study the parameterized complexity of SAT for “almost” βacyclic instances, using as parameter the formula’s distance from being βacyclic. As distance we use the size of smallest strong backdoor sets and the βhypertree width. As a byproduct we obtain the W[1]hardness of SAT parameterized by the (undirected) cliquewidth of the incidence graph, which disproves a conjecture by Fischer, Makowsky, and Ravve (Discr. Appl. Math. 156, 2008).
THE ENUMERATION OF VERTEX INDUCED SUBGRAPHS WITH RESPECT TO THE NUMBER OF COMPONENTS
, 2009
"... Inspired by the study of community structure in connection networks, we introduce the graph polynomial Q (G; x, y), the bivariate generating function which counts the number of connected components in induced subgraphs. We give a recursive definition of Q (G; x, y) using vertex deletion, vertex con ..."
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Inspired by the study of community structure in connection networks, we introduce the graph polynomial Q (G; x, y), the bivariate generating function which counts the number of connected components in induced subgraphs. We give a recursive definition of Q (G; x, y) using vertex deletion, vertex contraction and deletion of a vertex together with its neighborhood and prove a universality property. We relate Q (G; x, y) to other known graph invariants and graph polynomials, among them partition functions, the Tutte polynomial, the independence and matching polynomials, and the universal edge elimination polynomial introduced by I. Averbouch, B. Godlin and J.A. Makowsky (2008). We show that Q(G; x, y) is vertex reconstructible in the sense of Kelly and Ulam, discuss its use in computing residual connectedness reliability. Finally we show that the computation of Q(G; x, y) is ♯Phard, but Fixed Parameter
A most general edge elimination polynomial
 GraphTheoretic Concepts in Computer Science, 34nd International Workshop, WG 2006
, 2008
"... Abstract. We look for graph polynomials which satisfy recurrence relations on three kinds of edge elimination: edge deletion, edge contraction and edge extraction, i.e., deletion of edges together with their end points. Like in the case of deletion and contraction only (J.G. Oxley and D.J.A. Welsh 1 ..."
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Cited by 9 (1 self)
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Abstract. We look for graph polynomials which satisfy recurrence relations on three kinds of edge elimination: edge deletion, edge contraction and edge extraction, i.e., deletion of edges together with their end points. Like in the case of deletion and contraction only (J.G. Oxley and D.J.A. Welsh 1979), it turns out that there is a most general polynomial satisfying such recurrence relations, which we call ξ(G, x, y, z). We show that the new polynomial simultaneously generalizes the Tutte polynomial, the matching polynomial, and the recent generalization of the chromatic polynomial proposed by K.Dohmen, A.Pönitz and P.Tittman (2003), including also the independent set polynomial of I. Gutman and F. Harary, (1983) and the vertexcover polynomial of F.M. Dong, M.D. Hendy, K.T. Teo and C.H.C. Little (2002). We give three definitions of the new polynomial: first, the most general recursive definition, second, an explicit one, using a set expansion formula, and finally, a partition function, using counting of weighted graph homomorphisms. We prove the equivalence of the three definitions. Finally, we discuss the complexity of computing ξ(G, x, y, z). 1