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On Independent Sets and Bicliques in Graphs
"... Bicliques of graphs have been studied extensively, partially motivated by the large number of applications. One of the main algorithmic interests is in designing algorithms to enumerate all maximal bicliques of a (bipartite) graph. Polynomialtime reductions have been used explicitly or implicitly t ..."
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Bicliques of graphs have been studied extensively, partially motivated by the large number of applications. One of the main algorithmic interests is in designing algorithms to enumerate all maximal bicliques of a (bipartite) graph. Polynomialtime reductions have been used explicitly or implicitly to design polynomial delay algorithms to enumerate all maximal bicliques. Based on polynomialtime Turing reductions, various algorithmic problems on (maximal) bicliques can be studied by considering the related problem for (maximal) independent sets. In this line of research, we improve Prisner’s upper bound on the number of maximal bicliques [Combinatorica, 2000] and show that the maximum number of maximal bicliques in a graph on n vertices is exactly 3 n/3 (up to a polynomial factor). The main results of this paper are O(1.3642 n) time algorithms to compute the number of maximal independent sets and maximal bicliques in a graph.
An exact exponential time algorithm for counting bipartite cliques ∗
"... We present a simple exact algorithm for counting bicliques of given size in a bipartite graph on n vertices. We achieve running time of O(1.2491 n), improving upon known exact algorithms for finding and counting bipartite cliques. ..."
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We present a simple exact algorithm for counting bicliques of given size in a bipartite graph on n vertices. We achieve running time of O(1.2491 n), improving upon known exact algorithms for finding and counting bipartite cliques.
Computing #2SAT of Grids, GridCylinders and GridTori Boolean Formulas
"... We present an adaptation of transfer matrix method for signed grids, gridcylinders and gridtori. We use this adaptation to count the number of satisfying assignments of Boolean Formulas in 2CNF whose corresponding associated graph has such grid topologies. 1 ..."
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We present an adaptation of transfer matrix method for signed grids, gridcylinders and gridtori. We use this adaptation to count the number of satisfying assignments of Boolean Formulas in 2CNF whose corresponding associated graph has such grid topologies. 1
A Minimum Spanning Tree for the #2SAT Problem
"... Abstract. #2SAT is a classical #Pcomplete problem. We present here, a novel algorithm for given a 2CF Σ, to build a minimum spanning tree for its constraint graph GΣ assuming dynamic weights on the edges of the input graph. ..."
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Abstract. #2SAT is a classical #Pcomplete problem. We present here, a novel algorithm for given a 2CF Σ, to build a minimum spanning tree for its constraint graph GΣ assuming dynamic weights on the edges of the input graph.
A Note for Parametric Complexity of #2SAT
"... Abstract. We present some results about the parametric complexity for counting the number of truth assignments for two Conjunctive Forms (2CF’s), such problem is denoted as #2SAT. It is common to analyze the computational complexity for #2SAT regarding the number of variables or the number of claus ..."
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Abstract. We present some results about the parametric complexity for counting the number of truth assignments for two Conjunctive Forms (2CF’s), such problem is denoted as #2SAT. It is common to analyze the computational complexity for #2SAT regarding the number of variables or the number of clauses on the input formula F We consider here, the time complexity analysis for #2SAT based on a positive integer parameter k, wherek represents the number of satisfy assignments of F. We establish that for some values of k, thequestion: