Strict Cost Sharing Schemes for Steiner Forest

Gupta et al. [J. ACM, 54 (2007), article 11] and Gupta, Kumar, and Roughgarden [in Proceedings of the ACM Symposium on Theory of Computing, ACM, New York, 2003, pp. 365– 372] recently developed an elegant framework for the development of randomized approximation algorithms for rent-or-buy network design problems. The essential building block of this framework is an approximation algorithm for the underlying network design problem that admits a strict cost sharing scheme. Such cost sharing schemes have also proven to be useful in the development of approximation algorithms in the context of two-stage stochastic optimization with recourse. The main contribution of this paper is to show that the Steiner forest problem admits cost shares that are 3-strict and 4-group-strict. As a consequence, we derive surprisingly simple approximation algorithms for the multicommodity rent-or-buy and the multicast rent-or-buy problems with approximation ratios 5 and 6, improving over the previous best approximation ratios of 6.828 and 12.8, respectively. We also show that no approximation ratio better than 4.67 can be achieved using the sampleand-augment framework in combination with the currently best known Steiner forest approximation algorithms. In the context of two-stage stochastic optimization, our result leads to a 6-approximation algorithm for the stochastic Steiner tree problem in the black-box model and a 5-approximation algorithm for the stochastic Steiner forest problem in the independent decision model.