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

T.-H Hubert Chan,Haotian Jiang,Shaofeng H.-C Jiang

doi:10.1145/3378571

Abstract

We present a unified (randomized) 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 a tree 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.

Highlights

We study prize collecting versions of two important optimization problems: the prize collecting traveling salesman problem (PCTSP) and the prize collecting Steiner tree problem (PCSTP)
A solution for either problem is a connected subgraph with vertex set from the metric
The setting that we consider was suggested by Bienstock et al [8], and they used LP rounding to give a 2.5-approximation algorithm for the PCTSP and a 3-approximation for the PCSTP

Summary

Introduction

We study prize collecting versions of two important optimization problems: the prize collecting traveling salesman problem (PCTSP) and the prize collecting Steiner tree problem (PCSTP). We need to carefully define W2 and combine the solutions F1 and F2, because, as we remarked before, even if the approximate algorithm returns F1 for the instance W1, a near optimal global solution might not visit any terminals in W1. An issue in applying this framework is that after obtaining solutions F1 and F2 for the sub-instances, in the case that F1 and F2 are far away from each other as in our example in Figure 1 where it is too costly to connect them directly, it is not clear immediately which of F1 and F2 should be the weight part of the global solution and which would become the penalty part. Together with other sections, are omitted due to space limit, and they can be found in the full version [11]

Preliminaries

Overview

Sparsity Estimator for PCX

Decomposition into Sparse Instances