Elementary Approximation Algorithms for Prize Collecting Steiner Tree Problems

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 Vof size at least 2, a weight function w:E??+, and a penalty function p:T??+. The goal is to find a forest Fthat minimizes the cost of the edges of Fplus the penalties paid for subsets T i whose vertices are not all connected by F. Our main result is a combinatorial $(3-\frac{4}{n})$-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 ratio we achieve is better than that of the best known combinatorial algorithm for this problem, which is the 3-approximation of Sharma, Swamy, and Williamson [13]. Furthermore, our algorithm is obtained using an elegant application of the local ratio method and is much simpler and practical, since unlike the algorithm of Sharma et al., it does not use submodular function minimization. Our approach gives a $(2-\frac{1}{n-1})$-approximation for the prize collecting Steiner tree problem (all subsets T i are of size 2 and there is some root vertex rthat belongs to all of them). This latter algorithm is in fact the local ratio version of the primal-dual algorithm of Goemans and Williamson [7]. Another special case of our main algorithm is Bar-Yehuda's local ratio $(2-\frac{2}{n})$-approximation for the generalized Steiner forest problem (all the penalties are infinity) [3]. 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.