The notion of treewidth is of considerable interest in relation to NP-hard problems. Indeed, several studies have shown that the tree-decomposition method can be used to solve many basic optimization problems in polynomial time when treewidth is bounded, even if, for arbitrary graphs, computing the treewidth is NP-hard. Several papers present heuristics with computational experiments. For many graphs the discrepancy between the heuristic results and the best lower bounds is still very large. The aim of this paper is to propose two new methods for computing the treewidth of graphs: a heuristic and a metaheuristic. The heuristic returns good results in a short computation time, whereas the metaheuristic (a Tabu search method) returns the best results known to have been obtained so far for all the DIMACS vertex coloring / treewidth benchmarks (a well-known collection of graphs used for both vertex coloring and treewidth problems.) Our results actually improve on the previous best results for treewidth problems in 53% of the cases. Moreover, we identify properties of the triangulation process to optimize the computing time of our method.
@article{RO_2004__38_1_13_0,
author = {Clautiaux, Fran\c{c}ois and Moukrim, Aziz and N\`egre, St\'ephane and Carlier, Jacques},
title = {Heuristic and metaheuristic methods for computing graph treewidth},
journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
pages = {13--26},
publisher = {EDP-Sciences},
volume = {38},
number = {1},
year = {2004},
doi = {10.1051/ro:2004011},
mrnumber = {2083969},
zbl = {1092.90065},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/ro:2004011/}
}
TY - JOUR AU - Clautiaux, François AU - Moukrim, Aziz AU - Nègre, Stéphane AU - Carlier, Jacques TI - Heuristic and metaheuristic methods for computing graph treewidth JO - RAIRO - Operations Research - Recherche Opérationnelle PY - 2004 SP - 13 EP - 26 VL - 38 IS - 1 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ro:2004011/ DO - 10.1051/ro:2004011 LA - en ID - RO_2004__38_1_13_0 ER -
%0 Journal Article %A Clautiaux, François %A Moukrim, Aziz %A Nègre, Stéphane %A Carlier, Jacques %T Heuristic and metaheuristic methods for computing graph treewidth %J RAIRO - Operations Research - Recherche Opérationnelle %D 2004 %P 13-26 %V 38 %N 1 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/ro:2004011/ %R 10.1051/ro:2004011 %G en %F RO_2004__38_1_13_0
Clautiaux, François; Moukrim, Aziz; Nègre, Stéphane; Carlier, Jacques. Heuristic and metaheuristic methods for computing graph treewidth. RAIRO - Operations Research - Recherche Opérationnelle, Tome 38 (2004) no. 1, pp. 13-26. doi : 10.1051/ro:2004011. http://archive.numdam.org/articles/10.1051/ro:2004011/
[1] and, Local Search in Combinatorial Optimization. Series in Discrete Mathematics and Optimization. John Wiley and Sons (1997). | MR | Zbl
[2] ,, and, A survey on very large-scale neighborhood search techniques. Discrete Appl. Math. 123 (2002) 75-102. | MR | Zbl
[3] , and, Complexity of finding embeddings in a k-tree. SIAM J. Alg. Disc. Meth. 8 (1987) 277-284. | MR | Zbl
[4] and, Characterisation and recognition of partial 3-trees. SIAM J. Alg. Disc. Meth. 7 (1986) 305-314. | MR | Zbl
[5] , A linear time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 25 (1996) 1305-1317. | MR | Zbl
[6] and, A decomposition algorithm for network reliability evaluation. Discrete Appl. Math. 65 (1993) 141-156. | MR | Zbl
[7] ,, and, New lower and upper bounds for graph treewidth. WEA 2003, Lect. Notes Comput. Sci. 2647 (2003) 70-80. | MR | Zbl
[8] , Algorithms for minimum coloring, maximum clique, minimum coloring cliques and maximum independent set of a chordal graph. SIAM J. Comput. 1 (1972) 180-187. | MR | Zbl
[9] and, Tabu search. Kluwer Academic Publishers (1998). | MR | Zbl
[10] , Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York (1980). | MR | Zbl
[11] , and, Bayesian updating in causal probabilistic networks by local computations. Comput. Statist. Quaterly 4 (1990) 269-282. | MR | Zbl
[12] and, The Second DIMACS Implementation Challenge: NP-Hard Problems: Maximum Clique, Graph Coloring, and Satisfiability. Series in Discrete Math. Theor. Comput. Sci. Amer. Math. Soc. (1993).
[13] , Frequency Assignment, Models and Algorithms. Ph.D. Thesis, Universiteit Maastricht (1999).
[14] , and, Treewidth: Computational experiments. Fund. Inform. 49 (2001) 301-312. | MR
[15] , and, Evaluating network reliability and 2-edge-connected reliability in linear time for bounded pathwidth graphs. Algorithmica 27 (2000) 316-336. | MR | Zbl
[16] , and, Méthode de décomposition appliquée à la coloration de graphes, in ROADEF (2002).
[17] and, Graph minors. II. Algorithmic aspects of tree-width. J. Algorithms 7 (1986) 309-322. | MR | Zbl
[18] , Triangulated graphs and the elimination process. J. Math. Anal. Appl. 32 (1970) 597-609. | MR | Zbl
[19] , A graph-theoretic study of the numerical solution of sparse positive definite systems of linear equations, in Graph Theory and Computing, edited by R.C. Reed. Academic Press (1972) 183-217. | MR | Zbl
[20] , and, Algorithmic aspects of vertex elimination on graphs. SIAM J. Comput. 5 (1976) 146-160. | MR | Zbl
[21] and, Simple linear-time algorithm to test chordality of graphs, test acyclicity of hypergraphs, and selectivity reduce acyclic hypergraphs. SIAM J. Comput. 13 (1984) 566-579. | MR | Zbl
Cité par Sources :
