Abstract
We present hardness results, approximation heuristics, and exact algorithms for bottleneck labeled optimization problems arising in the context of graph theory. This long-established model partitions the set of edges into classes, each of which is identified by a unique color. The generic objective is to construct a subgraph of prescribed structure (such as that of being an s-t path, a spanning tree, or a perfect matching) while trying to avoid over-picking or under-picking edges from any given color.
Due to space limitations, some proofs were omitted from this extended abstract. We refer the reader to the full version of this paper (currently available online at http://www.lamsade.dauphine.fr/~monnot), in which all missing details are provided.
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
Ageev, A.A., Sviridenko, M.: Pipage rounding: A new method of constructing algorithms with proven performance guarantee. Journal of Combinatorial Optimization 8(3), 307–328 (2004)
Aissi, H., Bazgan, C., Vanderpooten, D.: Approximation complexity of min-max (regret) versions of shortest path, spanning tree, and knapsack. In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 862–873. Springer, Heidelberg (2005)
Aissi, H., Bazgan, C., Vanderpooten, D.: Complexity of the min-max (regret) versions of cut problems. In: Deng, X., Du, D.-Z. (eds.) ISAAC 2005. LNCS, vol. 3827, pp. 789–798. Springer, Heidelberg (2005)
Aissi, H., Bazgan, C., Vanderpooten, D.: Pseudo-polynomial algorithms for min-max and min-max regret problems. In: 5th ISORA, pp. 171–178 (2005)
Averbakh, I., Berman, O.: Categorized bottleneck-minisum path problems on networks. Operations Research Letters 16(5), 291–297 (1994)
Bertsimas, D., Sim, M.: Robust discrete optimization and network flows. Mathematical Programming Series B 98(1), 49–71 (2003)
Broersma, H., Li, X.: Spanning trees with many or few colors in edge-colored graphs. Discussiones Mathematicae Graph Theory 17(2), 259–269 (1997)
Broersma, H., Li, X., Woeginger, G., Zhang, S.: Paths and cycles in colored graphs. Australasian Journal on Combinatorics 31, 299–311 (2005)
Brüggemann, T., Monnot, J., Woeginger, G.: Local search for the minimum label spanning tree problem with bounded color classes. Operations Research Letters 31(3), 195–201 (2003)
Chang, R.-S., Leu, S.-J.: The minimum labeling spanning trees. Information Processing Letters 63(5), 277–282 (1997)
Ehrgott, M., Gandibleux, X. (eds.): Multiple Criteria Optimization: State of the Art Annotated Bibliographic Survey. International Series in Operations Research and Management Science, vol. 52. Kluwer Academic Publishers, Dordrecht (2002)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman and Company, New York (1979)
Hassin, R., Monnot, J., Segev, D.: Approximation algorithms and hardness results for labeled connectivity problems. In: Královič, R., Urzyczyn, P. (eds.) MFCS 2006. LNCS, vol. 4162, pp. 480–491. Springer, Heidelberg (2006)
Itai, A., Rodeh, M., Tanimoto, S.L.: Some matching problems for bipartite graphs. Journal of the ACM 25(4), 517–525 (1978)
Karger, D.R., Motwani, R., Ramkumar, G.D.S.: On approximating the longest path in a graph. Algorithmica 18(1), 82–98 (1997)
Karzanov, A.V.: Maximum matchings of given weight in complete and complete bipartite graphs. Kibernetika 1, 7–11 (1987), English translation in CYBNAW, 23, 8–13
Kouvelis, P., Yu, G.: Robust Discrete Optimization and its Applications. Kluwer Academic Publishers, Dordrecht (1997)
Lawler, E.L.: Combinatorial Optimization: Networks and Matroids. Holt, Rinehart and Winston, New York (1976)
Monnot, J.: The labeled perfect matching in bipartite graphs. Information Processing Letters 96(3), 81–88 (2005)
Papadimitriou, C.H., Yannakakis, M.: On the approximability of trade-offs and optimal access of web sources. In: 41st FOCS, pp. 86–92 (2000)
Punnen, A.P.: Traveling salesman problem under categorization. Operations Research Letters 12(2), 89–95 (1992)
Punnen, A.P.: On bottleneck assignment problems under categorization. Computers and Operations Research 31(1), 151–154 (2004)
Richey, M.B., Punnen, A.P.: Minimum perfect bipartite matchings and spanning trees under categorization. Discrete Applied Mathematics 39(2), 147–153 (1992)
Ulungu, E.L., Teghem, J.: Multi-objective combinatorial optimization problems: A survey. Journal of Multi-Criteria Decision Analysis 3, 83–104 (1994)
White, D.J.: A bibliography on the applications of mathematical programming multiple-objective methods. Journal of the Operational Research Society 41(8), 669–691 (1990)
Yi, T., Murty, K.G., Spera, C.: Matchings in colored bipartite networks. Discrete Applied Mathematics 121(1-3), 261–277 (2002)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Hassin, R., Monnot, J., Segev, D. (2007). The Complexity of Bottleneck Labeled Graph Problems. In: Brandstädt, A., Kratsch, D., Müller, H. (eds) Graph-Theoretic Concepts in Computer Science. WG 2007. Lecture Notes in Computer Science, vol 4769. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74839-7_31
Download citation
DOI: https://doi.org/10.1007/978-3-540-74839-7_31
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-74838-0
Online ISBN: 978-3-540-74839-7
eBook Packages: Computer ScienceComputer Science (R0)