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 inĀ [9], 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.
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References
Bogart, K.P., Fishburn, P.C., Isaak, G., Langley, L.: Proper and unit tolerance graphs. Discrete Applied MathematicsĀ 60(1-3), 99ā117 (1995)
Busch, A.H.: A characterization of triangle-free tolerance graphs. Discrete Applied MathematicsĀ 154(3), 471ā477 (2006)
Busch, A.H., Isaak, G.: Recognizing bipartite tolerance graphs in linear time. In: BrandstƤdt, A., Kratsch, D., MĆ¼ller, H. (eds.) WG 2007. LNCS, vol.Ā 4769, pp. 12ā20. Springer, Heidelberg (2007)
Diestel, R.: Graph Theory, 3rd edn. Springer, Berlin (2005)
Felsner, S.: Tolerance graphs and orders. Journal of Graph TheoryĀ 28, 129ā140 (1998)
Felsner, S., MĆ¼ller, R., Wernisch, L.: Trapezoid graphs and generalizations, geometry and algorithms. Discrete Applied MathematicsĀ 74, 13ā32 (1997)
Fishburn, P.C., Trotter, W.T.: Split semiorders. Discrete MathematicsĀ 195, 111ā126 (1999)
Fredman, M.L.: On computing the length of longest increasing subsequences. Discrete MathematicsĀ 11, 29ā35 (1975)
Golumbic, M.C., Monma, C.L.: A generalization of interval graphs with tolerances. In: Proceedings of the 13th Southeastern Conference on Combinatorics, Graph Theory and Computing, Congressus Numerantium, vol.Ā 35, pp. 321ā331 (1982)
Golumbic, M.C., Monma, C.L., Trotter, W.T.: Tolerance graphs. Discrete Applied MathematicsĀ 9(2), 157ā170 (1984)
Golumbic, M.C., Siani, A.: Coloring algorithms for tolerance graphs: Reasoning and scheduling with interval constraints. In: Joint International Conferences on Artificial Intelligence, Automated Reasoning, and Symbolic Computation (AISC/Calculemus), pp. 196ā207 (2002)
Golumbic, M., Trenk, A.: Tolerance Graphs. Cambridge Studies in Advanced Mathematics (2004)
Grƶtshcel, M., LovĆ”sz, L., Schrijver, A.: The Ellipsoid Method and its Consequences in Combinatorial Optimization. CombinatoricaĀ 1, 169ā197 (1981)
Hayward, R.B., Shamir, R.: A note on tolerance graph recognition. Discrete Applied MathematicsĀ 143(1-3), 307ā311 (2004)
Isaak, G., Nyman, K., Trenk, A.: A hierarchy of classes of bounded bitolerance orders. Ars CombinatoriaĀ 69 (2003)
Keil, J.M., Belleville, P.: Dominating the complements of bounded tolerance graphs and the complements of trapezoid graphs. Discrete Applied MathematicsĀ 140(1-3), 73ā89 (2004)
Langley, L.: Interval tolerance orders and dimension. PhD thesis, Dartmouth College (June 1993)
McKee, T., McMorris, F.: Topics in Intersection Graph Theory. Society for Industrial and Applied Mathematics. SIAM, Philadelphia (1999)
Mertzios, G.B., Sau, I., Zaks, S.: A New Intersection Model and Improved Algorithms for Tolerance Graphs. Technical report, RWTH Aachen University (March 2009)
Narasimhan, G., Manber, R.: Stability and chromatic number of tolerance graphs. Discrete Applied MathematicsĀ 36, 47ā56 (1992)
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Mertzios, G.B., Sau, I., Zaks, S. (2010). A New Intersection Model and Improved Algorithms for Tolerance Graphs. In: Paul, C., Habib, M. (eds) Graph-Theoretic Concepts in Computer Science. WG 2009. Lecture Notes in Computer Science, vol 5911. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11409-0_25
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DOI: https://doi.org/10.1007/978-3-642-11409-0_25
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