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The Maximum Clique Problem
, 1999
"... Contents 1 Introduction 2 1.1 Notations and Definitions . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Problem Formulations 4 2.1 Integer Programming Formulations . . . . . . . . . . . . . . . . . . . 5 2.2 Continuous Formulations . . . . . . . . . . . . . . . . . . . . . . . . 8 3 Computation ..."
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Cited by 140 (20 self)
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Contents 1 Introduction 2 1.1 Notations and Definitions . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Problem Formulations 4 2.1 Integer Programming Formulations . . . . . . . . . . . . . . . . . . . 5 2.2 Continuous Formulations . . . . . . . . . . . . . . . . . . . . . . . . 8 3 Computational Complexity 12 4 Bounds and Estimates 15 5 Exact Algorithms 19 5.1 Enumerative Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 19 5.2 Exact Algorithms for the Unweighted Case . . . . . . . . . . . . . . 21 5.3 Exact Algorithms for the Weighted Case . . . . . . . . . . . . . . . . 25 6 Heuristics 27 6.1 Sequential Greedy Heuristics . . . . . . . . . . . . . . . . . . . . . . 28 6.2 Local Search Heuristics . . . . . . . . . . . . . . . . . . . . . . . . . 29 6.3 Advanced Search Heuristics . . . . . . . . . . . . . . . . . . . . . . . 30 6.3.1 Simulated annealing . . . . . . . . . . . . . . . . . . . . . . . 30 6.3.2 Neural networks . . . . . . . . . . . . . . . . . . . . . . . .
Constructing Cliques Using Restricted Backtracking
, 1996
"... The restricted backtracking algorithmic paradigm is applied to the Maximum Clique Problem. The notion of backtracking coordinates is introduced. The program searches for those cliques whose backtracking coordinates are bounded by the values given in the input. ..."
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Cited by 15 (6 self)
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The restricted backtracking algorithmic paradigm is applied to the Maximum Clique Problem. The notion of backtracking coordinates is introduced. The program searches for those cliques whose backtracking coordinates are bounded by the values given in the input.
An Exact Parallel Algorithm For The Maximum Clique Problem
 In High Performance and Software in Nonlinear Optimization
, 1998
"... . In this paper we present a portable exact parallel algorithm for the maximum clique problem on general graphs. Computational results with random graphs and some test graphs from applications are presented. The algorithm is parallelized using the Message Passing Interface (MPI) standard. The algori ..."
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Cited by 3 (0 self)
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. In this paper we present a portable exact parallel algorithm for the maximum clique problem on general graphs. Computational results with random graphs and some test graphs from applications are presented. The algorithm is parallelized using the Message Passing Interface (MPI) standard. The algorithm is based on the CarraghanPardalos exact algorithm (for unweighted graphs) and incorporates a variant of the greedy randomized adaptive search procedure (GRASP) for maximum independent set of Feo, Resende, and Smith (1994) to obtain good starting solutions. 1. INTRODUCTION Let G= (V,E) be an undirected weighted graph where V = {v 1 , v 2 , . . . , v n } is the set of vertices in G, and E #V×V is the set of edges in G. Each vertex v i #V is associated with a positive weight w i . For a subset S #V , we define the weight of S to be W (S) =å i#S w i and G(S) = (S,E #S×S) as the subgraph induced by S. The size of the vertex set is throughout this paper denoted by n. The adjacenc...
A novel evolutionary formulation of the maximum independent set problem
 JOURNAL OF COMBINATORIAL OPTIMIZATION
, 2003
"... We introduce a novel evolutionary formulation of the problem of finding a maximum independent set of a graph. The new formulation is based on the relationship that exists between a graph’s independence number and its acyclic orientations. It views such orientations as individuals and evolves them wi ..."
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Cited by 3 (1 self)
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We introduce a novel evolutionary formulation of the problem of finding a maximum independent set of a graph. The new formulation is based on the relationship that exists between a graph’s independence number and its acyclic orientations. It views such orientations as individuals and evolves them with the aid of evolutionary operators that are very heavily based on the structure of the graph and its acyclic orientations. The resulting heuristic has been tested on some of the Second DIMACS Implementation Challenge benchmark graphs, and has been found to be competitive when compared to several of the other heuristics that have also been tested on those graphs.
Finding Maximum Cliques with Distributed Ants
 GECCO 2004, Lecture Notes in Computer Science
"... Abstract. In this paper we describe an ant system algorithm (ASMC) for the problem of finding the maximum clique in a given graph. In the algorithm each ant has only local knowledge of the graph. Working together the ants induce a candidate set of vertices from which a clique can be constructed. The ..."
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Cited by 2 (1 self)
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Abstract. In this paper we describe an ant system algorithm (ASMC) for the problem of finding the maximum clique in a given graph. In the algorithm each ant has only local knowledge of the graph. Working together the ants induce a candidate set of vertices from which a clique can be constructed. The algorithm was designed so that it can be easily implemented in a distributed system. One such implementation is also described in the paper. For 22 of the 30 graphs tested ASMC found the optimal solution. For the remaining graphs ASMC produced solutions that are within 16 % of the optimal, with most being within 8 % of the optimal. The performance of ASMC is comparable to existing algorithms. 1
A BranchandPrice Approach for the Maximum Weight Independent Set Problem
, 2005
"... The maximum weight independent set problem (MWISP) is one of the most wellknown and wellstudied problems in combinatorial optimization. This paper presents a novel approach to solve MWISP exactly by decomposing the original graph into vertexinduced subgraphs. The approach solves MWISP for the or ..."
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Cited by 1 (0 self)
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The maximum weight independent set problem (MWISP) is one of the most wellknown and wellstudied problems in combinatorial optimization. This paper presents a novel approach to solve MWISP exactly by decomposing the original graph into vertexinduced subgraphs. The approach solves MWISP for the original graph by solving MWISP on the subgraphs in order to generate columns for a branchandprice framework. The authors investigate different implementation techniques that can be associated with the approach and offer computational results to identify the strengths and weaknesses of each implementation technique.