Optimization problems in graphs with locational uncertainty

TL;DR

This paper studies spatial graph optimization under vertex location uncertainty, proving NP-hardness, and introduces exact and approximate algorithms for minimizing worst-case edge distances.

Contribution

It establishes NP-hardness for these problems and develops an exact algorithm with cutting planes, plus a conservative approximation linked to affine decision rules.

Findings

NP-hardness proven for spanning trees and s-t paths

Exact algorithm combining integer programming and cutting planes

Conservative approximation equivalent to affine decision rules

Abstract

Many discrete optimization problems amount to selecting a feasible set of edges of least weight. We consider in this paper the context of spatial graphs where the positions of the vertices are uncertain and belong to known uncertainty sets. The objective is to minimize the sum of the distances of the chosen set of edges for the worst positions of the vertices in their uncertainty sets. We first prove that these problems are $\cal NP$-hard even when the feasible sets consist either of all spanning trees or of all $s-t$ paths. Given this hardness, we propose an exact solution algorithm combining integer programming formulations with a cutting plane algorithm, identifying the cases where the separation problem can be solved efficiently. We also propose a conservative approximation and show its equivalence to the affine decision rule approximation in the context of Euclidean distances. We compare our algorithms to three deterministic reformulations on instances inspired by the scientific literature for the Steiner tree problem and a facility location problem.