Abstract
We study the optimal linear arrangement (OLA) problem on interval graphs. Several linear layout problems that are NP-hard on general graphs are solvable in polynomial time on interval graphs. We prove that, quite surprisingly, optimal linear arrangement of interval graphs is NP-hard. The same result holds for permutation graphs. We present a lower bound and a simple and fast 2-approximation algorithm based on any interval model of the input graph.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Biedl, T., Brejova, B., Demaine, E., Hamel, A., Lopez-Ortiz, A., Vinar, T.: Finding Hidden Independent Sets in Interval Graphs. Theoretical Computer Science 310(1-3), 287–307 (2004)
Brandstädt, A., Le, V.B., Spinrad, J.: Graph Classes: A Survey. In: SIAM Monographs on Discrete Math. Appl., vol. 3. SIAM, Philadelphia (1999)
Chung, F.R.K.: Labelings of graphs. In: Selected Topics in graph theory, pp. 151–168. Academic Press, San Diego (1988)
Díaz, J., Petit, J., Serna, M.J.: A survey of graph layout problems. ACM Computing Surveys 34(3), 313–356 (2002)
Even, G., Naor, S., Schieber, B., Rao, S., Even, G.: Divide-and-conquer approximation algorithms via spreading metrics. Journal of the ACM 47(4), 585–616 (2000)
Even, S., Shiloach, Y.: NP-Completeness of Several Arrangement Problems, Technical Report #43, Computer Science Dept., The Technion, Haifa, Israel (1975)
Goldberg, M.A., Klipker, I.A.: Minimal placing of trees on a line, Technical report, Physico-Technical Institute of Low Temperatures, Academy of Sciences of Ukranian SSR, USSR (1976)
Farach-Colton, M., Huang, Y., Woolford, J.L.L.: Discovering temporal relations in molecular pathways using protein-protein interactions. In: Proceedings of the 8th Annual International Conference on Computational Molecular Biology (RECOMB), San Diego, California, USA, March 27-31, pp. 150–156. ACM Press, New York (2004)
Feige, U.: Approximating the bandwidth via volume respecting embeddings. J. Comput. System Sci. 60, 510–539 (2000)
Garey, M.R., Graham, R.L., Johnson, D.S., Knuth, D.E.: Complexity results for bandwidth minimization. SIAM J. Appl. Math. 34, 477–495 (1978)
Garey, M.R., Johnson, D.S.: Computers and Intractability - A Guide to the Theory and Practice of NP-Completeness. W.H. Freeman, San Francisco (1979)
Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York (1980)
Johnson, D.: The NP-completeness column: an ongoing guide. J. Algorithms 6, 434–451 (1985)
Kleitman, D.J., Vohra, R.V.: Computing the bandwidth of interval graphs. SIAM J. Discrete Math. 3, 373–375 (1990)
Kratsch, D., Stewart, L.K.: Approximating bandwidth by mixing layouts of interval graphs. SIAM Journal on Discrete Mathematics 15, 435–449 (2002)
McConnell, R., Spinrad, J.: Modular decomposition and transitive orientation. Discrete Math. 201, 189–241 (1999)
Monien, B.: The bandwidth minimization problem for caterpillars with hair length 3 is NP-complete. SIAM J. Algebraic Discrete Methods 7, 505–512 (1986)
Rao, S., Richa, A.: New approximation techniques for some ordering problems. In: Proceedings of the Ninth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 1998), pp. 211–218. ACM, New York (1998)
Shiloach, Y.: A minimum linear arrangement algorithm for undirected trees. SIAM Journal on Computing 8, 15–32 (1979)
Sprague, A.P.: An O(n logn) algorithm for bandwidth of interval graphs. SIAM J. Discrete Math. 7, 213–220 (1994)
Unger, W.: The complexity of the approximation of the bandwidth problem. In: Proceedings of the Thirty-ninth Annual IEEE Symposium on Foundations of Computer Science, Palo Alto, CA (1998)
Waterman, M.: Introduction to computational biology: Maps, sequences and genomes. Chapman and Hall, Boca Raton (1995)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Cohen, J., Fomin, F., Heggernes, P., Kratsch, D., Kucherov, G. (2006). Optimal Linear Arrangement of Interval Graphs. In: Královič, R., Urzyczyn, P. (eds) Mathematical Foundations of Computer Science 2006. MFCS 2006. Lecture Notes in Computer Science, vol 4162. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11821069_24
Download citation
DOI: https://doi.org/10.1007/11821069_24
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-37791-7
Online ISBN: 978-3-540-37793-1
eBook Packages: Computer ScienceComputer Science (R0)