An LMP O(log n)-Approximation Algorithm for Node Weighted Prize Collecting Steiner Tree

In the node-weighted prize-collecting Steiner tree problem (NW-PCST) we are given an undirected graph G = (V, E), non-negative costs c(v) and penalties p(v) for each v. V. The goal is to find a tree T that minimizes the total cost of the vertices spanned by T plus the total penalty of vertices not in T. This problem is well-known to be set-cover hard to approximate. Moss and Rabani (STOC'01) presented a primal-dual Lagrangean-multiplier-preserving O(ln broken vertical bar V broken vertical bar)-approximation algorithm for this problem. We show a serious problem with the algorithm, and present a new, fundamentally different primal-dual method achieving the same performance guarantee. Our algorithm introduces several novel features to the primal-dual method that may be of independent interest.