# Approximation and Hardness Results for Label Cut and Related Problems

## Abstract

We investigate a natural combinatorial optimization problem called the *Label Cut* problem. Given an input graph *G* with a source *s* and a sink *t*, the edges of *G* are classified into different categories, represented by a set of *labels*. The labels may also have weights. We want to pick a subset of labels of minimum cardinality (or minimum total weight), such that the removal of all edges with these labels disconnects *s* and *t*. We give the first non-trivial approximation and hardness results for the Label Cut problem. Firstly, we present an \(O(\sqrt{m})\)-approximation algorithm for the Label Cut problem, where *m* is the number of edges in the input graph. Secondly, we show that it is **NP**-hard to approximate Label Cut within \(2^{\log ^{1 - 1/\log\log^c n} n}\) for any constant *c* < 1/2, where *n* is the input length of the problem. Thirdly, our techniques can be applied to other previously considered optimization problems. In particular we show that the Minimum Label Path problem has the same approximation hardness as that of Label Cut, simultaneously improving and unifying two known hardness results for this problem which were previously the best (but incomparable due to different complexity assumptions).

## Preview

Unable to display preview. Download preview PDF.

## References

- 1.Arora, S., Lund, C.: Hardness of Approximation. In: Hochbaum, D. (ed.) Approximation Algorithms for NP-hard Problems, pp. 399–446. PWS Publishing Company (1997)Google Scholar
- 2.Broersma, H., Li, X.: Spanning trees with many or few colors in edge-colored graphs. Discussiones Mathematicae Graph Theory 17(2), 259–269 (1997)zbMATHMathSciNetGoogle Scholar
- 3.Carr, R., Doddi, S., Konjevod, G., Marathe, M.: On the red-blue set cover problem. In: Proc. of SODA, pp. 345–353 (2000)Google Scholar
- 4.Coudert, D., Datta, P., Perennes, S., Rivano, H., Voge, M.-E.: Shared risk resource group: complexity and approximability issues. Parallel Processing Letters 17(2), 169–184 (2007)CrossRefMathSciNetGoogle Scholar
- 5.Dinur, I., Fischer, E., Kindler, G., Raz, R., Safra, S.: PCP characterizations of NP: towards a polynomially-small error-probability. In: Proc. of STOC, pp. 29–40 (1999)Google Scholar
- 6.Dinur, I., Safra, S.: On the hardness of approximating Label Cover. Information Processing Letters 89(5), 247–254 (2004)CrossRefMathSciNetGoogle Scholar
- 7.Hassin, R., Monnot, J., Segev, D.: Approximation algorithms and hardness results for labeled connectivity problems. Journal of Combinatorial Optimization 14(4), 437–453 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
- 8.Jha, S., Sheyner, O., Wing, J.M.: Two formal analyses of attack graphs. In: Proceedings of the 15th IEEE Computer Security Foundations Workshop, Nova Scotia, Canada, June 2002, pp. 49–63 (2002)Google Scholar
- 9.Khuller, S., Moss, A., Naor, J.: The budgeted maximum coverage problem. Information Processing Letters 70(1), 39–45 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
- 10.Sheyner, O., Wing, J.M.: Tools for Generating and Analyzing Attack Graphs. In: Proceedings of Workshop on Formal Methods for Components and Objects, pp. 344–371 (2004)Google Scholar
- 11.Sheyner, O., Haines, J., Jha, S., Lippmann, R., Wing, J.M.: Automated Generation and Analysis of Attack Graphs. In: Proceedings of the IEEE Symposium on Security and Privacy, Oakland, CA, May 2002, pp. 273–284 (2002)Google Scholar
- 12.Wirth, H.: Multicriteria Approximation of Network Design and Network Upgrade Problems. PhD thesis, Department of Computer Science, Würzburg University (2001)Google Scholar