A Unified PTAS for Prize Collecting TSP and Steiner Tree Problem in Doubling Metrics

TL;DR

This paper introduces a unified polynomial-time approximation scheme for prize collecting TSP and Steiner tree problems in doubling metrics, overcoming previous limitations by developing new techniques for divide-and-conquer strategies.

Contribution

It provides the first unified PTAS for these problems in doubling metrics, extending previous dynamic programming frameworks with novel techniques for handling penalties.

Findings

First unified PTAS for PCTSP and PCSTP in doubling metrics.

Overcomes the lack of QPTAS for these problems in such metric spaces.

Develops new divide-and-conquer techniques for penalty and edge length separation.

Abstract

We present a unified polynomial-time approximation scheme (PTAS) for the prize collecting traveling salesman problem (PCTSP) and the prize collecting Steiner tree problem (PCSTP) in doubling metrics. Given a metric space and a penalty function on a subset of points known as terminals, a solution is a subgraph on points in the metric space, whose cost is the weight of its edges plus the penalty due to terminals not covered by the subgraph. Under our unified framework, the solution subgraph needs to be Eulerian for PCTSP, while it needs to be connected for PCSTP. Before our work, even a QPTAS for the problems in doubling metrics is not known. Our unified PTAS is based on the previous dynamic programming frameworks proposed in [Talwar STOC 2004] and [Bartal, Gottlieb, Krauthgamer STOC 2012]. However, since it is unknown which part of the optimal cost is due to edge lengths and which part is due to penalties of uncovered terminals, we need to develop new techniques to apply previous divide-and-conquer strategies and sparse instance decompositions.