A note on the prize collecting traveling salesman problem
Mathematical Programming: Series A and B
When Trees Collide: An Approximation Algorithm for theGeneralized Steiner Problem on Networks
SIAM Journal on Computing
A General Approximation Technique for Constrained Forest Problems
SIAM Journal on Computing
The prize collecting Steiner tree problem: theory and practice
SODA '00 Proceedings of the eleventh annual ACM-SIAM symposium on Discrete algorithms
A unified approach to approximating resource allocation and scheduling
Journal of the ACM (JACM)
Local ratio: A unified framework for approximation algorithms. In Memoriam: Shimon Even 1935-2004
ACM Computing Surveys (CSUR)
A group-strategyproof mechanism for Steiner forests
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Network design for information networks
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
On the Equivalence between the Primal-Dual Schema and the Local Ratio Technique
SIAM Journal on Discrete Mathematics
The prize-collecting generalized steiner tree problem via a new approach of primal-dual schema
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
An efficient cost-sharing mechanism for the prize-collecting Steiner forest problem
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Approximation algorithms for prize collecting forest problems with submodular penalty functions
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Prize-collecting steiner networks via iterative rounding
LATIN'10 Proceedings of the 9th Latin American conference on Theoretical Informatics
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This paper deals with approximation algorithms for the prize collecting generalized Steiner forest problem, defined as follows. The input is an undirected graph G=(V,E), a collection T={T"1,...,T"k}, each a subset of V of size at least 2, a weight function w:E-R^+, and a penalty function p:T-R^+. The goal is to find a forest F that minimizes the cost of the edges of F plus the penalties paid for subsets T"i whose vertices are not all connected by F. Our main result is a (3-4n)-approximation for the prize collecting generalized Steiner forest problem, where n=2 is the number of vertices in the graph. This obviously implies the same approximation for the special case called the prize collecting Steiner forest problem (all subsets T"i are of size 2). The approximation algorithm is obtained by applying the local ratio method, and is much simpler than the best known combinatorial algorithm for this problem. Our approach gives a (2-1n-1)-approximation for the prize collecting Steiner tree problem (all subsets T"i are of size 2 and there is some root vertex r that belongs to all of them). This latter algorithm is in fact the local ratio version of the primal-dual algorithm of Goemans and Williamson [M.X. Goemans, D.P. Williamson, A general approximation technique for constrained forest problems, SIAM Journal on Computing 24 (2) (April 1995) 296-317]. Another special case of our main algorithm is Bar-Yehuda's local ratio (2-2n)-approximation for the generalized Steiner forest problem (all the penalties are infinity) [R. Bar-Yehuda, One for the price of two: a unified approach for approximating covering problems, Algorithmica 27 (2) (June 2000) 131-144]. Thus, an important contribution of this paper is in providing a natural generalization of the framework presented by Goemans and Williamson, and later by Bar-Yehuda.