Abstract
The notion of tree-decomposition has very strong theoretical interest related to NP-Hard problems. Indeed, several studies show that it can be used to solve many basic optimization problems in polynomial time when the treewidth is bounded. So, given an arbitrary graph, its decomposition and its treewidth have to be determined, but computing the treewidth of a graph is NP-Hard. Hence, several papers present heuristics with computational experiments, but for many instances of graphs, the heuristic results are far from the best lower bounds.
The aim of this paper is to propose new lower and upper bounds for the treewidth. We tested them on the well known DIMACS benchmark for graph coloring, so we can compare our results to the best bounds of the literature. We improve the best lower bounds dramatically, and our heuristic method computes good bounds within a very small computing time.
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References
The second dimacs implementation challenge: NP-hard problems: Maximum clique, graph coloring, and satisfiability, 1992–1993.
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.
—, Necessary edges in k-chordalisation of graphs, Tech. Report UU-CS-2000-27, Institute for Information and Computing Sciences, Utrecht University, 2000.
H. Bodlaender and R. Möhring, The pathwidth and treewidth of cographs, SIAM J. Disc. Math. 6 (1993), 181–188.
J. Carlier and C. Lucet, A decomposition algorithm for network reliability evaluation, Discrete Applied Mathematics 65 (1993), 141–156.
F. Clautiaux, A. Moukrim, S. Nègre, and J. Carlier, A tabu search minimising upper bounds for graph treewidth, Report, 2003.
D. Fulkerson and O. Gross, Incidence matrices and interval graphs, Pacific J. Math. 15 (1965), 835–855.
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. Jensen, S. Lauritzen, and K. Olesen, Bayesian updating in causal probabilistic networks by local computations, Computational Statistics Quaterly 4 (1990), 269–282.
A. Koster, Frequency assignment, models and algorithms, Ph.D. thesis, Universiteit Maastricht, 1999.
A. Koster, H. Bodlaender, and S. van Hoesel, Treewidth: Computational experiments, Fundamenta Informaticae 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, Algorithmica 27 (2000), 316–336.
C. Lucet, F. Mendes, and A. Moukrim, Méthode de décomposition appliquée à la coloration de graphes, ROADEF, 2002.
N. Robertson and P. Seymour, Graph minors. ii algorithmic aspects of tree-width, Journal of Algorithms 7 (1986), 309–322.
D. Rose, Triangulated graphs and the elimination process, J Math. Anal. Appl. 32 (1970), 597–609.
—, A graph-theoretic study of the numerical solution of sparse positive definite systems of linear equations, Graph Theory and Computing (R. C. Reed, ed.), Academic Press (1972), 183–217.
D. Rose, E. Tarjan, and G. Lueker, Algorithmic aspects of vertex elimination on graphs, SIAM J. Comput. 5 (1976), 146–160.
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Clautiaux, F., Carlier, J., Moukrim, A., Nègre, S. (2003). New Lower and Upper Bounds for Graph Treewidth. In: Jansen, K., Margraf, M., Mastrolilli, M., Rolim, J.D.P. (eds) Experimental and Efficient Algorithms. WEA 2003. Lecture Notes in Computer Science, vol 2647. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44867-5_6
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DOI: https://doi.org/10.1007/3-540-44867-5_6
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