ISAAC 2008: Algorithms and Computation pp 16-27

# The Complexity of Minimum Convex Coloring

• Frank Kammer
• Torsten Tholey
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5369)

## Abstract

A coloring of the vertices of a graph is called convex if each subgraph induced by all vertices of the same color is connected. We consider three variants of recoloring a colored graph with minimal cost such that the resulting coloring is convex. Two variants of the problem are shown to be $${\mathcal{NP}}$$-hard on trees even if in the initial coloring each color is used to color only a bounded number of vertices. For graphs of bounded treewidth, we present a polynomial-time (2 + ε)-approximation algorithm for these two variants and a polynomial-time algorithm for the third variant. Our results also show that, unless $${\mathcal{NP}} \subseteq DTIME(n^{O(\log \log n)})$$, there is no polynomial-time approximation algorithm with a ratio of size (1 − o(1))ln ln n for the following problem: Given pairs of vertices in an undirected graph of bounded treewidth, determine the minimal possible number l for which all except l pairs can be connected by disjoint paths.

## Keywords

Convex Coloring Maximum Disjoint Paths Problem

## References

1. 1.
Alon, N., Moshkovitz, D., Safra, S.: Algorithmic construction of sets for k-restrictions. ACM Transactions on Algorithms 2, 153–177 (2006)
2. 2.
Andrews, M., Zhang, L.: Hardness of the undirected edge-disjoint paths problem. In: Proc. 37th Annual ACM Symposium on Theory of Computing (STOC 2005), pp. 276–283 (2005)Google Scholar
3. 3.
Bar-Yehuda, R., Feldman, I., Rawitz, D.: Improved approximation algorithm for convex recoloring of trees. In: Erlebach, T., Persinao, G. (eds.) WAOA 2005. LNCS, vol. 3879, pp. 55–68. Springer, Heidelberg (2006)
4. 4.
Bodlaender, H.L.: A partial k-arboretum of graphs with bounded tree width. Theoret. Comput. Sci. 209, 1–45 (1998)
5. 5.
Bodlaender, H.L., Kloks, T.: Efficient and constructive algorithms for the pathwidth and treewidth of graphs. J. Algorithms 21, 358–402 (1996)
6. 6.
Chen, X., Hu, X., Shuai, T.: Inapproximability and approximability of maximal tree routing and coloring. J. Comb. Optim. 11, 219–229 (2006)
7. 7.
Feige, U.: A threshold of ln n for approximating set cover. J. ACM 45, 634–652 (1998)
8. 8.
Karp, R.M.: On the computational complexity of combinatorial problems. Networks 5, 45–68 (1975)
9. 9.
Lynch, J.F.: The equivalence of theorem proving and the interconnection problem. (ACM) SIGDA Newsletter 5, 31–36 (1975)
10. 10.
Moran, S., Snir, S.: Convex recolorings of strings and trees: definitions, hardness results and algorithms. In: Dehne, F., López-Ortiz, A., Sack, J.-R. (eds.) WADS 2005. LNCS, vol. 3608, pp. 218–232. Springer, Heidelberg (2005)
11. 11.
Moran, S., Snir, S.: Efficient approximation of convex recoloring. J. Comput. System Sci. 73, 1078–1089 (2007)
12. 12.
Robertson, N., Seymour, P.D.: Graph minors. I. Excluding a forest. J. Comb. Theory Ser. B 35, 39–61 (1983)
13. 13.
Snir, S.: Computational Issues in Phylogenetic Reconstruction: Analytic Maximum Likelihood Solutions, and Convex Recoloring. Ph.D. Thesis, Department of Computer Science, Technion, Haifa, Israel (2004)Google Scholar