Abstract
Multiple interval graphs are variants of interval graphs where instead of a single interval, each vertex is assigned a set of intervals on the real line. We study the complexity of the MAXIMUM CLIQUE problem in several classes of multiple interval graphs. The MAXIMUM CLIQUE problem, or the problem of finding the size of the maximum clique, is known to be NP-complete for t-interval graphs when t ≥ 3 and polynomial-time solvable when t = 1. The problem is also known to be NP-complete in t-track graphs when t ≥ 4 and polynomial-time solvable when t ≤ 2. We show that MAXIMUM CLIQUE is already NP-complete for unit 2-interval graphs and unit 3-track graphs. Further, we show that the problem is APX-complete for 2-interval graphs, 3-track graphs, unit 3-interval graphs and unit 4-track graphs. We also introduce two new classes of graphs called t-circular interval graphs and t-circular track graphs and study the complexity of the MAXIMUM CLIQUE problem in them. On the positive side, we present a polynomial time t-approximation algorithm for WEIGHTED MAXIMUM CLIQUE on t-interval graphs, improving earlier work with approximation ratio 4t.
This work was partially supported by the grant ANR-09-JCJC-0041.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Alon, N.: Piercing d-intervals. Discrete and Computational Geometry 19, 333–334 (1998)
Asinowski, A., Cohen, E., Golumbic, M.C., Limouzy, V., Lipshteyn, M., Stern, M.: Vertex Intersection Graphs of Paths on a Grid. Journal of Graph Algorithms and Applications 16(2), 129–150 (2012)
Aumann, Y., Lewenstein, M., Melamud, O., Pinter, R.Y., Yakhini, Z.: Dotted interval graphs and high throughput genotyping. In: Proc. of the 16th Annual Symposium on Discrete Algorithms, SODA 2005, pp. 339–348 (2005)
Bar-Yehuda, R., Halldórsson, M.M., Naor, J.S., Shachnai, H., Shapira, I.: Scheduling split intervals. SIAM J. Comput. 36, 1–15 (2006)
Berman, P., Fujito, T.: On Approximation Properties of the Independent Set Problem for Degree 3 Graphs. In: Sack, J.-R., Akl, S.G., Dehne, F., Santoro, N. (eds.) WADS 1995. LNCS, vol. 955, pp. 449–460. Springer, Heidelberg (1995)
Butman, A., Hermelin, D., Lewenstein, M., Rawitz, D.: Optimization problems in multiple-interval graphs. In: Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2007, pp. 268–277 (2007)
Cabello, S., Cardinal, J., Langerman, S.: The Clique Problem in Ray Intersection Graph. arXiv (November 2011), http://arxiv.org/pdf/1111.5986.pdf
Chlebík, M., Chlebíková, J.: The complexity of combinatorial optimization problems on d-dimensional boxes. SIAM Journal on Discrete Mathematics 21(1), 158–169 (2007)
Crochemore, M., Hermelin, D., Landau, G.M., Vialette, S.: Approximating the 2-Interval Pattern Problem. In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 426–437. Springer, Heidelberg (2005)
Garey, M.R., Johnson, D.S.: Rectilinear steiner tree problem is NP-complete. SIAM J. Appl. Math. 6, 826–834 (1977)
Gavril, F.: Algorithms for a maximum clique and a maximum independent set of a circle graph. Networks 3, 261–273 (1973)
Gavril, F.: Maximum weight independent sets and cliques in intersection graphs of filaments. Information Processing Letters 73(56), 181–188 (2000)
Hochbaum, D.S., Levin, A.: Cyclical scheduling and multi-shift scheduling: Complexity and approximation algorithms. Disc. Optimiz. 3(4), 327–340 (2006)
Hsu, W.-L.: Maximum weight clique algorithms for circular-arc graphs and circle graphs. SIAM J. Comput. 14(1), 224–231 (1985)
Jiang, M.: Clique in 3-track interval graphs is APX-hard. arXiv (April 2012), http://arxiv.org/pdf/1204.2202v1.pdf
Jiang, M., Zhang, Y.: Parameterized Complexity in Multiple-Interval Graphs: Domination, Partition, Separation, Irredundancy. arXiv (October 2011), http://arxiv.org/pdf/1110.0187v1.pdf
Kaiser, T.: Transversals of d-Intervals. Discrete Comput. Geom. 18, 195–203 (1997)
Kammer, F., Tholey, T., Voepel, H.: Approximation Algorithms for Intersection Graphs. In: Serna, M., Shaltiel, R., Jansen, K., Rolim, J. (eds.) APPROX 2010, LNCS, vol. 6302, pp. 260–273. Springer, Heidelberg (2010)
König, F.G.: Sorting with objectives. PhD thesis, Technische Universität Berlin (2009)
Kratochvíl, J., Nešetřil, J.: Independent set and clique problems in intersection-defined classes of graphs. Commentationes Mathematicae Universitatis Carolinae 31, 85–93 (1990)
Middendorf, M., Pfeiffer, F.: The max clique problem in classes of string-graphs. Discrete Mathematics 108, 365–372 (1992)
Papadimitriou, C.H., Yannakakis, M.: Optimization, approximation, and complexity classes. J. Comput. System Sci. 43, 425–440 (1991)
Scheinerman, E.R.: The maximum interval number of graphs with given genus. Journal of Graph Theory 11(3), 441–446 (1987)
Scheinerman, E.R., West, D.B.: The interval number of a planar graph: Three intervals suffice. Journal of Combinatorial Theory, Series B 35(3), 224–239 (1983)
Trotter, W.T., Harary, F.: On double and multiple interval graphs. Journal of Graph Theory 3(3), 205–211 (1979)
Valiant, L.G.: Universality considerations in VLSI circuits. IEEE Transactions on Computers 30(2), 135–140 (1981)
West, D.B., Shmoys, D.B.: Recognizing graphs with fixed interval number is NP-complete. Discrete Applied Mathematics 8(3), 295–305 (1984)
Zuckerman, D.: Linear degree extractors and the inapproximability of max clique and chromatic number. In: Proc. 38th ACM Symp. Theory of Computing, STOC 2006, pp. 681–690 (2006)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Francis, M.C., Gonçalves, D., Ochem, P. (2012). The Maximum Clique Problem in Multiple Interval Graphs (Extended Abstract). In: Golumbic, M.C., Stern, M., Levy, A., Morgenstern, G. (eds) Graph-Theoretic Concepts in Computer Science. WG 2012. Lecture Notes in Computer Science, vol 7551. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34611-8_9
Download citation
DOI: https://doi.org/10.1007/978-3-642-34611-8_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-34610-1
Online ISBN: 978-3-642-34611-8
eBook Packages: Computer ScienceComputer Science (R0)