NDLI: A New Intersection Model and Improved Algorithms for Tolerance Graphs

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Zaks, Shmuel Mertzios, George B. Sau, Ignasi
Copyright Year	2009
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. 321331], 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 $\mathcal{O}(n\log n)$ algorithms for computing a minimum coloring and a maximum clique and an $\mathcal{O}(n^{2})$ algorithm for computing a maximum weight independent set in a tolerance graph with n vertices, thus improving the best known running times $\mathcal{O}(n^{2})$ and $\mathcal{O}(n^{3})$ for these problems, respectively.
Starting Page	1800
Ending Page	1813
Page Count	14
File Format	PDF
ISSN	08954801
DOI	10.1137/09075994X
e-ISSN	10957146
Journal	SIAM Journal on Discrete Mathematics (SJDMEC)
Issue Number	4
Volume Number	23
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2009-12-04
Access Restriction	Subscribed
Subject Keyword	tolerance graphs minimum coloring parallelogram graphs Dominating sets, independent sets, cliques intersection model Graph theory Graph algorithms maximum clique Graph representations Coloring of graphs and hypergraphs maximum weight independent set
Content Type	Text
Resource Type	Article
Subject	Mathematics

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