Primal–dual approximation algorithms for the prize-collecting Steiner tree problem

The primal–dual scheme has been used to provide approximation algorithms for many problems. Goemans and Williamson gave a (2-1/(n-1))-approximation for the Prize-Collecting Steiner Tree Problem that runs in O(n logn) time – it applies the primal–dual scheme once for each of the n vertices of the graph. We present a primal–dual algorithm that runs in O(n logn), as it applies this scheme only once, and achieves the slightly better ratio of (2-2/n). We also show a tight example for the analysis of the algorithm and discuss briefly a couple of other algorithms described in the literature.