Approximating optimization problems in graphs with locational uncertainty

TL;DR

This paper develops polynomial-time approximation algorithms for graph optimization problems with vertex location uncertainty, analyzing their approximation ratios and providing a specialized scheme for s-t path problems.

Contribution

It introduces two approximation algorithms for uncertain vertex location problems, including a PTS for s-t paths, with detailed analysis of their approximation ratios.

Findings

First algorithm uses a deterministic approximation with max distances.

Approximation ratio depends on subgraph structure and metric space properties.

A fully-polynomial scheme is provided for s-t path problems.

Abstract

Many combinatorial optimization problems can be formulated as the search for a subgraph that satisfies certain properties and minimizes the total weight. We assume here that the vertices correspond to points in a metric space and can take any position in given uncertainty sets. Then, the cost function to be minimized is the sum of the distances for the worst positions of the vertices in their uncertainty sets. We propose two types of polynomial-time approximation algorithms. The first one relies on solving a deterministic counterpart of the problem where the uncertain distances are replaced with maximum pairwise distances. We study in details the resulting approximation ratio, which depends on the structure of the feasible subgraphs and whether the metric space is Ptolemaic or not. The second algorithm is a fully-polynomial time approximation scheme for the special case of $s-t$ paths.