International Workshop on Combinatorial Algorithms

IWOCA 2014: Combinatorial Algorithms pp 13-23

# Approximation and Hardness Results for the Maximum Edges in Transitive Closure Problem

• Guillaume Blin
• Alexandru Popa
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8986)

## Abstract

In this paper we study the following problem, named Maximum Edges in Transitive Closure, which has applications in computational biology. Given a simple, undirected graph $$G = (V,E)$$ and a coloring of the vertices, remove a collection of edges from the graph such that each connected component is colorful (i.e., it does not contain two identically colored vertices) and the number of edges in the transitive closure of the graph is maximized.

The problem is known to be APX-hard, and no approximation algorithms are known for it. We improve the hardness result by showing that the problem is NP-hard to approximate within a factor of $$|V|^{1/3 - \varepsilon }$$, for any constant $$\varepsilon > 0$$. Additionally, we show that the problem is APX-hard already for the case when the number of vertex colors equals 3. We complement these results by showing the first approximation algorithm for the problem, with approximation factor $$\sqrt{2 \cdot \text {OPT}}$$.

## References

1. 1.
Adamaszek, A., Popa, A.: Algorithmic and hardness results for the colorful components problems. In: Pardo, A., Viola, A. (eds.) LATIN 2014. LNCS, vol. 8392, pp. 683–694. Springer, Heidelberg (2014)
2. 2.
Avidor, A., Langberg, M.: The multi-multiway cut problem. Theoret. Comput. Sci. 377(1–3), 35–42 (2007)
3. 3.
Bruckner, S., Hüffner, F., Komusiewicz, C., Niedermeier, R.: Evaluation of ILP-based approaches for partitioning into colorful components. In: Demetrescu, C., Marchetti-Spaccamela, A., Bonifaci, V. (eds.) SEA 2013. LNCS, vol. 7933, pp. 176–187. Springer, Heidelberg (2013)
4. 4.
Bruckner, S., Hüffner, F., Komusiewicz, C., Niedermeier, R., Thiel, S., Uhlmann, J.: Partitioning into colorful components by minimum edge deletions. In: Kärkkäinen, J., Stoye, J. (eds.) CPM 2012. LNCS, vol. 7354, pp. 56–69. Springer, Heidelberg (2012)
5. 5.
He, G., Liu, J., Zhao, C.: Approximation algorithms for some graph partitioning problems. J. Graph Algorithms Appl. 4(2), 1–11 (2000)
6. 6.
Petrank, E.: The hardness of approximation: gap location. Comput. Complex. 4(2), 133–157 (1994)
7. 7.
Rizzi, R., Sikora, F.: Some results on more flexible versions of graph motif. In: Hirsch, E.A., Karhumäki, J., Lepistö, A., Prilutskii, M. (eds.) CSR 2012. LNCS, vol. 7353, pp. 278–289. Springer, Heidelberg (2012)
8. 8.
Sankoff, D.: OMG! orthologs for multiple genomes - competing formulations. In: Chen, J., Wang, J., Zelikovsky, A. (eds.) ISBRA 2011. LNCS, vol. 6674, pp. 2–3. Springer, Heidelberg (2011)
9. 9.
Savard, O.T., Swenson, K.M.: A graph-theoretic approach for inparalog detection. BMC Bioinform. 13(S–19), S16 (2012)
10. 10.
Zheng, C., Swenson, K., Lyons, E., Sankoff, D.: OMG! orthologs in multiple genomes – competing graph-theoretical formulations. In: Przytycka, T.M., Sagot, M.-F. (eds.) WABI 2011. LNCS, vol. 6833, pp. 364–375. Springer, Heidelberg (2011)
11. 11.
Zuckerman, D.: Linear degree extractors and the inapproximability of max clique and chromatic number. Theory Comput. 3(1), 103–128 (2007)

© Springer International Publishing Switzerland 2015

## Authors and Affiliations

• 1
• Guillaume Blin
• 2
• 3
• Alexandru Popa
• 4
Email author
1. 1.Max-Planck-Institut Für InformatikSaarbrückenGermany
2. 2.LaBRI, UMR 5800University of BordeauxTalenceFrance
3. 3.CNRS, LaBRI, UMR 5800TalenceFrance
4. 4.Faculty of InformaticsMasaryk UniversityBrnoCzech Republic