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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.
Partitioning based algorithms for some colouring problems
 In Recent Advances in Constraints, volume 3978 of LNAI
, 2005
"... Abstract. We discuss four variants of the graph colouring problem, and present algorithms for solving them. The problems are kColourability, Max Ind kCOL, Max Val kCOL, and, finally, Max kCOL, which is the unweighted case of the Max kCut problem. The algorithms are based on the idea of partitio ..."
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Abstract. We discuss four variants of the graph colouring problem, and present algorithms for solving them. The problems are kColourability, Max Ind kCOL, Max Val kCOL, and, finally, Max kCOL, which is the unweighted case of the Max kCut problem. The algorithms are based on the idea of partitioning the domain of the problems into disjoint subsets, and then considering all possible instances were the variables are restricted to values from these partitions. If a pair of variables have been restricted to different partitions, then the constraint between them is always satisfied since the only allowed constraint is disequality. 1
A microstructure based approach to constraint satisfaction optimisation problems
 Recent Advances in Artificial Intelligience. Proceedings of the Eighteenth International Florida Artificial Intelligence Research Society Conference (FLAIRS2005
, 2005
"... We study two constraint satisfaction optimisation problems: The MAX VALUE problem for CSPs, which, somewhat simplified, aims at maximising the sum of the (weighted) variable values in the solution, and the MAX IND problem, where the goal is to find a satisfiable subinstance of the original instance ..."
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We study two constraint satisfaction optimisation problems: The MAX VALUE problem for CSPs, which, somewhat simplified, aims at maximising the sum of the (weighted) variable values in the solution, and the MAX IND problem, where the goal is to find a satisfiable subinstance of the original instance containing as many variables as possible. Both problems are NPhard to approximate within n 1−ɛ,ɛ> 0, where n is the number of variables in the problems, which implies that it is of interest to find exact algorithms. By exploiting properties of the microstructure, we construct algorithms for solving instances of these problems with small domain sizes, and then, using a probabilistic reasoning, we show how to get algorithms for more general versions of the problems. The resulting algorithms have running times of O ((0.585d) n) for MAX VALUE (d, 2)CSP, and O ((0.503d) n) for MAX IND (d, 2)CSP. Both algorithms represent the best known theoretical bounds for their respective problem, and, more importantly, the methods used are applicable to a wide range of optimisation problems.
An Approximation Algorithm for #kSAT
"... We present a simple randomized algorithm that approximates the number of satisfying assignments of Boolean formulas in conjunctive normal form. To the best of our knowledge this is the first algorithm which approximates #kSAT for any k ≥ 3 within a running time that is not only nontrivial, but als ..."
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We present a simple randomized algorithm that approximates the number of satisfying assignments of Boolean formulas in conjunctive normal form. To the best of our knowledge this is the first algorithm which approximates #kSAT for any k ≥ 3 within a running time that is not only nontrivial, but also significantly better than that of the currently fastest exact algorithms for the problem. More precisely, our algorithm is a randomized approximation scheme whose running time depends polynomially on the error tolerance and is mildly exponential in the number n of variables of the input formula. For example, even stipulating subexponentially small error tolerance, the number of solutions to 3CNF input formulas can be approximated in time O(1.5366 n). For 4CNF input the bound increases to O(1.6155 n). We further show how to obtain upper and lower bounds on the number of solutions to a CNF formula in a controllable way. Relaxing the requirements on the quality of the approximation, on kCNF input we obtain significantly reduced running times in comparison to the above bounds.