Heuristic and metaheuristic methods for computing graph treewidth
RAIRO - Operations Research - Recherche Opérationnelle, Tome 38 (2004) no. 1, pp. 13-26.

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.

DOI : https://doi.org/10.1051/ro:2004011
Mots clés : treewidth, elimination orderings, triangulated graphs, heuristic, metaheuristic, computational experiments
@article{RO_2004__38_1_13_0,
author = {Clautiaux, Fran\c cois 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},
zbl = {1092.90065},
mrnumber = {2083969},
language = {en},
url = {www.numdam.org/item/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://www.numdam.org/item/RO_2004__38_1_13_0/`

[1] E. Aarts and J.K. Lenstra, Local Search in Combinatorial Optimization. Series in Discrete Mathematics and Optimization. John Wiley and Sons (1997). | MR 1458630 | Zbl 0869.00019

[2] R. Ahuja, O. Ergun, J. Orlin and A. Punnen, A survey on very large-scale neighborhood search techniques. Discrete Appl. Math. 123 (2002) 75-102. | MR 1922331 | Zbl 1014.68052

[3] S. Arnborg, D.G. Corneil and A. Proskurowski, Complexity of finding embeddings in a k-tree. SIAM J. Alg. Disc. Meth. 8 (1987) 277-284. | MR 881187 | Zbl 0611.05022

[4] S. Arnborg and A. Proskurowki, Characterisation and recognition of partial 3-trees. SIAM J. Alg. Disc. Meth. 7 (1986) 305-314. | MR 830649 | Zbl 0597.05027

[5] H. Bodlaender, A linear time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 25 (1996) 1305-1317. | MR 1417901 | Zbl 0864.68074

[6] J. Carlier and C. Lucet, A decomposition algorithm for network reliability evaluation. Discrete Appl. Math. 65 (1993) 141-156. | MR 1380072 | Zbl 0848.90058

[7] F. Clautiaux, J. Carlier, A. Moukrim and S. Nègre, New lower and upper bounds for graph treewidth. WEA 2003, Lect. Notes Comput. Sci. 2647 (2003) 70-80. | MR 2051951 | Zbl 1023.68645

[8] F. Gavril, 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 327580 | Zbl 0227.05116

[9] F. Glover and M. Laguna, Tabu search. Kluwer Academic Publishers (1998). | MR 1665424 | Zbl 0930.90083

[10] M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York (1980). | MR 562306 | Zbl 0541.05054

[11] F. Jensen, S. Lauritzen and K. Olesen, Bayesian updating in causal probabilistic networks by local computations. Comput. Statist. Quaterly 4 (1990) 269-282. | MR 1073446 | Zbl 0715.68076

[12] D.S. Johnson and M.A. Trick, 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] A. Koster, Frequency Assignment, Models and Algorithms. Ph.D. Thesis, Universiteit Maastricht (1999).

[14] A. Koster, H. Bodlaender and S. Van Hoesel, Treewidth: Computational experiments. Fund. Inform. 49 (2001) 301-312. | MR 2154596

[15] C. Lucet, J.F. Manouvrier and J. Carlier, Evaluating network reliability and 2-edge-connected reliability in linear time for bounded pathwidth graphs. Algorithmica 27 (2000) 316-336. | MR 1759753 | Zbl 0971.68009

[16] C. Lucet, F. Mendes and A. Moukrim, Méthode de décomposition appliquée à la coloration de graphes, in ROADEF (2002).

[17] N. Robertson and P. Seymour, Graph minors. II. Algorithmic aspects of tree-width. J. Algorithms 7 (1986) 309-322. | MR 855559 | Zbl 0611.05017

[18] D. Rose, Triangulated graphs and the elimination process. J. Math. Anal. Appl. 32 (1970) 597-609. | MR 270957 | Zbl 0216.02602

[19] D. Rose, 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 341833 | Zbl 0266.65028

[20] D. Rose, R. Tarjan and G. Lueker, Algorithmic aspects of vertex elimination on graphs. SIAM J. Comput. 5 (1976) 146-160. | MR 408312 | Zbl 0353.65019

[21] R. Tarjan and M. Yannakakis, 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 749707 | Zbl 0545.68062