Improved Approximation Algorithms for the Spanning Star Forest Problem
A star graph is a tree of diameter at most two. A star forest is a graph that consists of node-disjoint star graphs. In the spanning star forest problem, given an unweighted graph G, the objective is to find a star forest that contains all vertices of G and has the maximum number of edges. This problem is the complement of the dominating set problem in the following sense: On a graph with n vertices, the size of the maximum spanning star forest is equal to n minus the size of the minimum dominating set.
We present a 0.71-approximation algorithm for this problem, improving upon the approximation factor of 0.6 of Nguyen et al. (SIAM J. Comput. 38:946–962, 2008). We also present a 0.64-approximation algorithm for the problem on node-weighted graphs. Finally, we present improved hardness of approximation results for the weighted (both edge-weighted and node-weighted) versions of the problem.
Our algorithms use a non-linear rounding scheme, which might be of independent interest.
KeywordsApproximation algorithms Spanning star forest problem
We thank Aravind Srinivasan for clarifying the status of the approximability of the weighted set cover problem.
- 1.Agra, A., Cardoso, D., Cerdeira, O., Rocha, E.: A spanning star forest model for the diversity problem in automobile industry. In: ECCO XVIII, Minsk, May 2005 Google Scholar
- 3.Athanassopoulos, S., Caragiannis, I., Kaklamanis, C., Kyropoulou, M.: An improved approximation bound for spanning star forest and color saving. In: Mathematical Foundations of Computer Science, pp. 90–101 (2009) Google Scholar
- 7.Goemans, M.: Mathematical programming and approximation algorithms. Lecture at Udine School, Udine, Italy (1996) Google Scholar
- 13.Slavík, P.: A tight analysis of the greedy algorithm for set cover. In: ACM Symposium on Theory of Computing, pp. 435–441 (1996) Google Scholar
- 14.Srinivasan, A.: New approaches to covering and packing problems. In: ACM-SIAM Symposium on Discrete Algorithms, pp. 567–576 (2001) Google Scholar
- 15.Srinivasan, A.: Personal communication, August 2009 Google Scholar