IWOCA 2016: Combinatorial Algorithms pp 281-292

# Linear Time Algorithms for Happy Vertex Coloring Problems for Trees

• N. R. Aravind
• Subrahmanyam Kalyanasundaram
• Anjeneya Swami Kare
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9843)

## Abstract

Given an undirected graph $$G = (V, E)$$ with $$|V| = n$$ and a vertex coloring, a vertex v is happy if v and all its neighbors have the same color. An edge is happy if its end vertices have the same color. Given a partial coloring of the vertices of the graph using k colors, the Maximum Happy Vertices (also called k-MHV) problem asks to color the remaining vertices such that the number of happy vertices is maximized. The Maximum Happy Edges (also called k-MHE) problem asks to color the remaining vertices such that the number of happy edges is maximized. For arbitrary graphs, k-MHV and k-MHE are NP-Hard for $$k \ge 3$$. In this paper we study these problems for trees. For a fixed k we present linear time algorithms for both the problems. In general, for any k the proposed algorithms take $$O(nk \log k)$$ and O(nk) time respectively.

## Keywords

Happy vertex Happy edge Graph coloring Coloring trees

## Notes

### Acknowledgement

We thank the anonymous reviewers for their detailed reviews and suggestions.

## References

1. 1.
Chopra, S., Rao, M.R.: On the multiway cut polyhedron. Networks 21(1), 51–89 (1991)
2. 2.
Dahlhaus, E., Johnson, D.S., Papadimitriou, C.H., Seymour, P.D., Yannakakis, M.: The complexity of multiway cuts (extended abstract). In: Proceedings of the Twenty-fourth Annual ACM Symposium on Theory of Computing, STOC 1992, pp. 241–251 (1992)Google Scholar
3. 3.
Deng, X., Lin, B., Zhang, C.: Multi-multiway cut problem on graphs of bounded branch width. In: Fellows, M., Tan, X., Zhu, B. (eds.) FAW-AAIM 2013. LNCS, vol. 7924, pp. 315–324. Springer, Heidelberg (2013)
4. 4.
Easley, D., Kleinberg, J.: Networks, Crowds, and Markets: Reasoning About a Highly Connected World. Cambridge University Press, New York (2010)
5. 5.
Even, S., Tarjan, R.E.: Network flow and testing graph connectivity. SIAM J. Comput. 4(4), 507–518 (1975)
6. 6.
Iwata, S., Fleischer, L., Fujishige, S.: A combinatorial strongly polynomial algorithm for minimizing submodular functions. J. ACM 48(4), 761–777 (2001)
7. 7.
Langberg, M., Rabani, Y., Swamy, C.: Approximation algorithms for graph homomorphism problems. In: Díaz, J., Jansen, K., Rolim, J.D.P., Zwick, U. (eds.) APPROX 2006 and RANDOM 2006. LNCS, vol. 4110, pp. 176–187. Springer, Heidelberg (2006)
8. 8.
Robertson, N., Seymour, P.: Graph minors. X. Obstructions to tree-decomposition. J. Comb. Theory, Ser. B 52(2), 153–190 (1991)
9. 9.
Zhang, P., Jiang, T., Li, A.: Improved approximation algorithms for the maximum happy vertices and edges problems. In: Xu, D., Du, D., Du, D. (eds.) COCOON 2015. LNCS, vol. 9198, pp. 159–170. Springer, Heidelberg (2015)
10. 10.
Zhang, P., Li, A.: Algorithmic aspects of homophyly of networks. Theor. Comput. Sci. 593, 117–131 (2015)

© Springer International Publishing Switzerland 2016

## Authors and Affiliations

• N. R. Aravind
• 1
• Subrahmanyam Kalyanasundaram
• 1
• Anjeneya Swami Kare
• 1
Email author
1. 1.Department of Computer Science and Engineering Indian Institute of TechnologyHyderabadIndia