# Heuristic and metaheuristic methods for computing graph treewidth

François Clautiaux; Aziz Moukrim; Stéphane Nègre; Jacques Carlier

RAIRO - Operations Research (2010)

- Volume: 38, Issue: 1, page 13-26
- ISSN: 0399-0559

## Access Full Article

top## Abstract

top## How to cite

topClautiaux, François, et al. "Heuristic and metaheuristic methods for computing graph treewidth ." RAIRO - Operations Research 38.1 (2010): 13-26. <http://eudml.org/doc/105299>.

@article{Clautiaux2010,

abstract = {
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.
},

author = {Clautiaux, François, Moukrim, Aziz, Nègre, Stéphane, Carlier, Jacques},

journal = {RAIRO - Operations Research},

keywords = {Treewidth; elimination orderings; triangulated graphs; heuristic; metaheuristic; computational experiments. ; treewidth; computational experiments},

language = {eng},

month = {3},

number = {1},

pages = {13-26},

publisher = {EDP Sciences},

title = {Heuristic and metaheuristic methods for computing graph treewidth },

url = {http://eudml.org/doc/105299},

volume = {38},

year = {2010},

}

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

DA - 2010/3//

PB - EDP Sciences

VL - 38

IS - 1

SP - 13

EP - 26

AB -
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.

LA - eng

KW - Treewidth; elimination orderings; triangulated graphs; heuristic; metaheuristic; computational experiments. ; treewidth; computational experiments

UR - http://eudml.org/doc/105299

ER -

## References

top- E. Aarts and J.K. Lenstra, Local Search in Combinatorial Optimization. Series in Discrete Mathematics and Optimization. John Wiley and Sons (1997).
- 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.
- 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.
- S. Arnborg and A. Proskurowki, Characterisation and recognition of partial 3-trees. SIAM J. Alg. Disc. Meth.7 (1986) 305-314.
- H. Bodlaender, A linear time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput.25 (1996) 1305-1317.
- J. Carlier and C. Lucet, A decomposition algorithm for network reliability evaluation. Discrete Appl. Math.65 (1993) 141-156.
- 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.
- 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.
- F. Glover and M. Laguna, Tabu search. Kluwer Academic Publishers (1998).
- M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York (1980).
- F. Jensen, S. Lauritzen and K. Olesen, Bayesian updating in causal probabilistic networks by local computations. Comput. Statist. Quaterly4 (1990) 269-282.
- 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).
- A. Koster, Frequency Assignment, Models and Algorithms. Ph.D. Thesis, Universiteit Maastricht (1999).
- A. Koster, H. Bodlaender and S. van Hoesel, Treewidth: Computational experiments. Fund. Inform.49 (2001) 301-312.
- C. Lucet, J.F. Manouvrier and J. Carlier, Evaluating network reliability and 2-edge-connected reliability in linear time for bounded pathwidth graphs. Algorithmica27 (2000) 316-336.
- C. Lucet, F. Mendes and A. Moukrim, Méthode de décomposition appliquée à la coloration de graphes, in ROADEF (2002).
- N. Robertson and P. Seymour, Graph minors. II. Algorithmic aspects of tree-width. J. Algorithms7 (1986) 309-322.
- D. Rose, Triangulated graphs and the elimination process. J. Math. Anal. Appl.32 (1970) 597-609.
- 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.
- D. Rose, R. Tarjan and G. Lueker, Algorithmic aspects of vertex elimination on graphs. SIAM J. Comput.5 (1976) 146-160.
- 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.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.