Euclidean Prize-collecting Steiner Forest

TL;DR

This paper develops simplified and new PTAS algorithms for Euclidean Steiner forest and its prize-collecting variants, highlighting structural properties and limitations in the general case.

Contribution

It provides a simpler analysis of existing PTAS for Euclidean Steiner forest and introduces a PTAS for the multiplicative prize-collecting case, with insights into computational hardness.

Findings

A new structural property for Euclidean Steiner forest

A modified dynamic programming approach for PTAS

Hardness results for general Euclidean prize-collecting Steiner forest

Abstract

In this paper, we consider Steiner forest and its generalizations, prize-collecting Steiner forest and k-Steiner forest, when the vertices of the input graph are points in the Euclidean plane and the lengths are Euclidean distances. First, we present a simpler analysis of the polynomial-time approximation scheme (PTAS) of Borradaile et al. [12] for the Euclidean Steiner forest problem. This is done by proving a new structural property and modifying the dynamic programming by adding a new piece of information to each dynamic programming state. Next we develop a PTAS for a well-motivated case, i.e., the multiplicative case, of prize-collecting and budgeted Steiner forest. The ideas used in the algorithm may have applications in design of a broad class of bicriteria PTASs. At the end, we demonstrate why PTASs for these problems can be hard in the general Euclidean case (and thus for PTASs we cannot go beyond the multiplicative case).