Improved Approximation Algorithms for Prize-Collecting Steiner Tree and TSP

We study the prize-collecting Steiner tree (PCST), prize-collecting traveling salesman (PCTSP), and prize-collecting path (PC-Path) problems. Given a graph $(V,E)$ with a cost on each edge and a penalty (a.k.a. prize) on each node, the goal is to find a tree (for PCST), cycle (for PCTSP), or path (for PC-Path) that minimizes the sum of the edge costs in the tree/cycle/path 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 by AT&T to the optimization of real-world telecommunications networks. The most recent improvements for the first two problems, a 2-approximation algorithm for each, appeared first in 1992; a 2-approximation for PC-Path appeared in 2003. The natural linear programming relaxation of PCST has an integrality gap of 2, which has been a barrier to further improvements for this problem. We present $(2-\epsilon)$-approximation algorithms for all three problems, connected by a unified technique for improving prize-collecting algorithms that allows us to circumvent the integrality gap barrier. Specifically, our approximation ratio for prize-collecting Steiner tree is below 1.9672.