# Improved Methods for Approximating Node Weighted Steiner Trees and Connected Dominating Sets

• Sudipto Guha
• Samir Khuller
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1530)

## Abstract

In this paper we study the Steiner tree problem in graphs for the case when vertices as well as edges have weights associated with them. A greedy approximation algorithm based on “spider decompositions” was developed by Klein and Ravi for this problem. This algorithm provides a worst case approximation ratio of 2 ln k, where k is the number of terminals. However, the best known lower bound on the approximation ratio is (1 − o(1))ln k, assuming that $$NP \not\subseteq DTIME[n^{O(\log \log n)}]$$, by a reduction from set cover.

We show that for the unweighted case we can obtain an approximation factor of ln k. For the weighted case we develop a new decomposition theorem, and generalize the notion of “spiders” to “branch-spiders”, that are used to design a new algorithm with a worst case approximation factor of 1.5 ln k. We then generalize the method to yield an approximation factor of (1.35 + ε) ln k, for any constant ε> 0. These algorithms, although polynomial, are not very practical due to their high running time; since we need to repeatedly find many minimum weight matchings in each iteration. We also develop a simple greedy algorithm that is practical and has a worst case approximation factor of 1.6103 ln k. The techniques developed for this algorithm imply a method of approximating node weighted network design problems defined by 0-1 proper functions as well.

These new ideas also lead to improved approximation guarantees for the problem of finding a minimum node weighted connected dominating set. The previous best approximation guarantee for this problem was 3 ln n due to Guha and Khuller. By a direct application of the methods developed in this paper we are able to develop an algorithm with an approximation factor of (1.35 + ε) ln n for any fixed ε> 0.

## Keywords

Minimum Weight Steiner Tree Approximation Factor Connected Subgraph Steiner Tree Problem
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
Berman, P., Ramaiyer, V.: Improved approximation algorithms for the Steiner tree problem. J. Algorithms 17, 381–408 (1994)
2. 2.
Bern, M., Plassmann, P.: The Steiner problem with edge lengths 1 and 2. Information Processing Letters 32, 171–176 (1989)
3. 3.
Cormen, T., Leiserson, C., Rivest, R.: Introduction to Algorithms. MIT Press, Cambridge (1989)Google Scholar
4. 4.
Feige, U.: A threshold of ln n for approximating set-cover. In: 28th ACM Symposium on Theory of Computing, pp. 314–318 (1996)Google Scholar
5. 5.
Garey, M.R., Johnson, D.S.: Computers and Intractability: A guide to the theory of NP-completeness. Freeman, San Francisco (1978)Google Scholar
6. 6.
Goemans, M.X., Williamson, D.P.: A general approximation technique for constrained forest problems. SIAM Journal on Computing 24, 296–317 (1995)
7. 7.
Guha, S., Khuller, S.: Approximation algorithms for connected dominating sets. Algorithmica 20, 374–387 (1998)
8. 8.
Hwang, F.K., Richards, D.S., Winter, P.: The Steiner tree problem. Annals of Discrete Mathematics, vol. 53. North-Holland, Amsterdam (1992)
9. 9.
Kou, L., Markowsky, G., Berman, L.: A fast algorithm for Steiner trees. Acta Informatica 15, 141–145 (1981)
10. 10.
Klein, P.N., Ravi, R.: A nearly best-possible approximation algorithm for node-weighted Steiner trees. J. Algorithms 19(1), 104–114 (1995)
11. 11.
Karpinsky, M., Zelikovsky, A.: New approximation algorithms for the Steiner tree problem. Journal of Combinatorial Optimization 1(1), 47–66 (1997)
12. 12.
Lund, C., Yannakakis, M.: On the hardness of approximating minimization problems. Journal of the ACM 41(5), 960–981 (1994)
13. 13.
Rayward-Smith, V.J.: The computation of nearly minimal Steiner trees in graphs. Internat. J. Math. Educ. Sci. Tech. 14, 15–23 (1983)
14. 14.
Takahashi, H., Matsuyama, A.: An approximate solution for the Steiner problem in graphs. Math. Japonica 24, 573–577 (1980)
15. 15.
Zelikovsky, A.: An 11/6 approx algo for the network Steiner problem. Algorithmica 9, 463–470 (1993)