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 [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 Mathematics 60(1-3), 99–117 (1995)
Busch, A.H.: A characterization of triangle-free tolerance graphs. Discrete Applied Mathematics 154(3), 471–477 (2006)
Busch, A.H., Isaak, G.: Recognizing bipartite tolerance graphs in linear time. In: Brandstädt, A., Kratsch, D., Müller, H. (eds.) WG 2007. LNCS, vol. 4769, pp. 12–20. Springer, Heidelberg (2007)
Diestel, R.: Graph Theory, 3rd edn. Springer, Berlin (2005)
Felsner, S.: Tolerance graphs and orders. Journal of Graph Theory 28, 129–140 (1998)
Felsner, S., Müller, R., Wernisch, L.: Trapezoid graphs and generalizations, geometry and algorithms. Discrete Applied Mathematics 74, 13–32 (1997)
Fishburn, P.C., Trotter, W.T.: Split semiorders. Discrete Mathematics 195, 111–126 (1999)
Fredman, M.L.: On computing the length of longest increasing subsequences. Discrete Mathematics 11, 29–35 (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. 35, pp. 321–331 (1982)
Golumbic, M.C., Monma, C.L., Trotter, W.T.: Tolerance graphs. Discrete Applied Mathematics 9(2), 157–170 (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. 196–207 (2002)
Golumbic, M., Trenk, A.: Tolerance Graphs. Cambridge Studies in Advanced Mathematics (2004)
Grötshcel, M., Lovász, L., Schrijver, A.: The Ellipsoid Method and its Consequences in Combinatorial Optimization. Combinatorica 1, 169–197 (1981)
Hayward, R.B., Shamir, R.: A note on tolerance graph recognition. Discrete Applied Mathematics 143(1-3), 307–311 (2004)
Isaak, G., Nyman, K., Trenk, A.: A hierarchy of classes of bounded bitolerance orders. Ars Combinatoria 69 (2003)
Keil, J.M., Belleville, P.: Dominating the complements of bounded tolerance graphs and the complements of trapezoid graphs. Discrete Applied Mathematics 140(1-3), 73–89 (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 Mathematics 36, 47–56 (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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