# Approximation algorithms for connected dominating sets

## Abstract

The dominating set problem in graphs asks for a minimum size subset of vertices with the following property: each vertex is required to either be in the dominating set, or adjacent to at least one node in the dominating set. We focus on the question of finding a *connected dominating set* of minimum size, where the graph induced by vertices in the dominating set is required to be *connected*. This problem arises in network testing, as well as in wireless communication.

Two polynomial time algorithms that achieve approximation factors of *O(H(Δ))* are presented, where *Δ* is the maximum degree, and *H* is the harmonic function. This question also arises in relation to the traveling tourist problem, where one is looking for the shortest tour such that each vertex is either visited, or has one of its neighbors visited. We study a generalization of the problem when the vertices have weights, and give an algorithm which achieves a performance ratio of 3 ln *n*. We also consider the more general problem of finding a connected dominating set of a specified set of vertices and provide a 3 ln *n* approximation factor. To prove the bound we also develop an optimal approximation algorithm for the unit node weighted Steiner tree problem.

## Keywords

Steiner Tree Steiner Tree Problem White Neighbor Marked Vertex Steiner Vertex## Preview

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## References

- 1.E. Arkin, M. Halldórsson and R. Hassin, “Approximating the tree and tour covers of a graph”,
*Information Processing Letters*, 47: 275–282, (1993).Google Scholar - 2.P. Berman, personal communication, May (1996).Google Scholar
- 3.P. Berman and M. Fürer, personal communication, May (1996).Google Scholar
- 4.T. Cormen, C. Leiserson, and R. Rivest,
*Introduction to Algorithms*, The MIT Press, 1989.Google Scholar - 5.U. Feige, “A threshold of ln
*n*for approximating set-cover”,*28th ACM Symposium on Theory of Computing*, pages 314–318, (1996).Google Scholar - 6.M. R. Garey and D. S. Johnson, “Computers and Intractability: A guide to the theory of NP-completeness”,
*Freeman*,*San Francisco*(1978).Google Scholar - 7.A. Kothari and V. Bharghavan, “Algorithms for unicast and multicast routing in adhoc networks”, manuscript.Google Scholar
- 8.P. N. Klein and R. Ravi, “A nearly best-possible approximation algorithm for node-weighted Steiner trees”,
*J. Algorithms*, 19(1):104–114, (1995).Google Scholar - 9.D. Kleitman and D. West, “Spanning trees with many leaves”,
*SIAM Journal on Discrete Mathematics*, 4(1):99–106, (1991).Google Scholar - 10.M. Karpinsky and A. Zelikovsky, “New Approximation Algorithms for the Steiner Tree Problems”,
*Technical report, Electronic Colloquium on Computational Complexity*(ECCC): TR95-030, (1995).Google Scholar - 11.C. Lund and M. Yannakakis, “On the hardness of approximating minimization problems”,
*Journal of the ACM*, 41(5): 960–981, (1994).Google Scholar - 12.C. S. Mata and J. S. B. Mitchell “Approximation algorithms for geometric tour and network design problems”,
*Proc. of the 11th Annual Symp. on Computational Geometry*, pages 360–369, (1995).Google Scholar - 13.S. Paul and R. Miller, “Locating faults in a systematic manner in a large heterogeneous network”,
*IEEE INFOCOM*, pages 522–529, (1995).Google Scholar - 14.C. Savage, “Depth-First search and the vertex cover problem”,
*Information Processing Letters*, 14(5): 233–235, (1982).Google Scholar - 15.P. Slavík “A Tight Analysis of the Greedy Algorithm for Set Cover”,
*28th ACM Symposium on Theory of Computing*, pages 435–441, (1996).Google Scholar