A Technique for Improving Approximation Algorithms for Prize-Collecting Problems

We study the prize-collecting versions of the Steiner tree (PCST) and traveling salesman (PCTSP) problems: given a graph (V,E) with costs on each edge and a penalty on each node, the goal is to find a tree (for PCST) or cycle (for PCTSP), that minimizes the sum of the edge costs in the tree/cycle and the penalties of the nodes not spanned by it. In addition to being a useful theoretical tool for helping to solve other optimization problems, PCST has been applied fruitfully to the optimization of real-world telecommunications networks. The most recent improvements for these problems, giving a 2-approximation algorithm for each, appeared first in 1992. The natural linear programming (LP) relaxation of PCST has an integrality ratio of 2, which has been a barrier to further improvements for this problem. We present (2 )-approximation algorithms for both problems, connected by a general technique for improving prize-collecting algorithms that manages to circumvent the integrality ratio barrier.