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A New Intersection Model and Improved Algorithms for Tolerance Graphs

Mertzios, G.B.; Sau, I.; Zaks, S.

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Authors

I. Sau

S. Zaks



Abstract

Tolerance graphs model interval relations in such a way that intervals can tolerate a certain degree of overlap without being in conflict. This class of graphs, which generalizes in a natural way both interval and permutation graphs, has attracted many research efforts since their introduction in [M. C. Golumbic and C. L. Monma, Congr. Numer., 35 (1982), pp. 321–331], as it finds many important applications in constraint-based temporal reasoning, resource allocation, and scheduling problems, among others. In this article we propose the first non-trivial intersection model for general tolerance graphs, given by three-dimensional parallelepipeds, which extends the widely known intersection model of parallelograms in the plane that characterizes the class of bounded tolerance graphs. Apart from being important on its own, this new representation also enables us to improve the time complexity of three problems on tolerance graphs. Namely, we present optimal O(n log n) algorithms for computing a minimum coloring and a maximum clique and an O(n2) algorithm for computing a maximum weight independent set in a tolerance graph with n vertices, thus improving the best known running times O(n2) and O(n3) for these problems, respectively.

Citation

Mertzios, G., Sau, I., & Zaks, S. (2010). A New Intersection Model and Improved Algorithms for Tolerance Graphs. SIAM Journal on Discrete Mathematics, 23(4), 1800-1813. https://doi.org/10.1137/09075994x

Journal Article Type Article
Publication Date Jan 1, 2010
Deposit Date Sep 5, 2014
Publicly Available Date Mar 29, 2024
Journal SIAM Journal on Discrete Mathematics
Print ISSN 0895-4801
Electronic ISSN 1095-7146
Publisher Society for Industrial and Applied Mathematics
Peer Reviewed Peer Reviewed
Volume 23
Issue 4
Pages 1800-1813
DOI https://doi.org/10.1137/09075994x
Keywords Tolerance graphs, Parallelogram graphs, Intersection model, Minimum coloring, Maximum clique, Maximum weight independent set.
Related Public URLs http://epubs.siam.org/doi/abs/10.1137/09075994X

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© 2009 Society for Industrial and Applied Mathematics. Unauthorized version of this article is prohibited.





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