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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 . . . . . . . . . . . . . . . . . . . . . . . .
A Column Generation Approach For Graph Coloring
 INFORMS Journal on Computing
, 1995
"... We present a method for solving the independent set formulation of the graph coloring problem (where there is one variable for each independent set in the graph). We use a column generation method for implicit optimization of the linear program at each node of the branchandbound tree. This approac ..."
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Cited by 73 (2 self)
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We present a method for solving the independent set formulation of the graph coloring problem (where there is one variable for each independent set in the graph). We use a column generation method for implicit optimization of the linear program at each node of the branchandbound tree. This approach, while requiring the solution of a difficult subproblem as well as needing sophisticated branching rules, solves small to moderate size problems quickly. We have also implemented an exact graph coloring algorithm based on DSATUR for comparison. Implementation details and computational experience are presented. 1 INTRODUCTION The graph coloring problem is one of the most useful models in graph theory. This problem has been used to solve problems in school timetabling [10], computer register allocation [7, 8], electronic bandwidth allocation [11], and many other areas. These applications suggest that effective algorithms for solving the graph coloring problem would be of great importance. D...
A New Algorithm For The MaximumWeight Clique Problem
"... Given a graph, in the maximum clique problem one wants to find the largest number of vertices, any two of which are adjacent. In the maximumweight clique problem, the vertices have positive, integer weights, and one wants to find a clique with maximum weight. A recent algorithm for the maximum cliq ..."
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Cited by 19 (0 self)
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Given a graph, in the maximum clique problem one wants to find the largest number of vertices, any two of which are adjacent. In the maximumweight clique problem, the vertices have positive, integer weights, and one wants to find a clique with maximum weight. A recent algorithm for the maximum clique problem is here used as a basis for developing an algorithm for the weighted case. Computational experiments with random graphs show that this new algorithm is faster than earlier algorithms in many cases. A set of weighted graphs obtained from the problem of constructing good constant weight errorcorrecting codes are proposed as test cases for maximumweight clique algorithms
APPROXIMATING MAXIMUM STABLE SET AND MINIMUM GRAPH COLORING PROBLEMS WITH THE POSITIVE SEMIDEFINITE RELAXATION
"... We compute approximate solutions to the maximum stable set problem and the minimum graph coloring problem using a positive semidefinite relaxation. The positive semidefinite programs are solved using an implementation of the dual scaling algorithm that takes advantage of the sparsity inherent in m ..."
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Cited by 9 (1 self)
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We compute approximate solutions to the maximum stable set problem and the minimum graph coloring problem using a positive semidefinite relaxation. The positive semidefinite programs are solved using an implementation of the dual scaling algorithm that takes advantage of the sparsity inherent in most graphs and the structure inherent in the problem formulation. From the solution to the relaxation, we apply a randomized algorithm to find approximate maximum stable sets and a modification of a popular heuristic to find graph colorings. We obtained high quality answers for graphs with over 1000 vertices and almost 7000 edges.
A BRANCHANDPRICE APPROACH FOR Graph Multicoloring
"... We present a branchandprice framework for solving the graph multicoloring problem. We propose column generation to implicitly optimize the linear programming relaxation of an independent set formulation (where there is one variable for each independent set in the graph) for graph multicoloring. T ..."
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Cited by 2 (0 self)
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We present a branchandprice framework for solving the graph multicoloring problem. We propose column generation to implicitly optimize the linear programming relaxation of an independent set formulation (where there is one variable for each independent set in the graph) for graph multicoloring. This approach, while requiring the solution of a difficult subproblem, is a promising method to obtain good solutions for small to moderate size problems quickly. Some implementation details and initial computational experience are presented.
Cliques in kpartite Graphs and their Application in Textile Engineering
"... In many practical cases one has to choose an arrangement of di#erent ..."
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Cited by 1 (0 self)
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In many practical cases one has to choose an arrangement of di#erent