Article Source

• Optimization problems in graphs with locational uncertainty

Optimization problems in graphs with locational uncertainty

• Speaker: Michaël Poss
• Le Laboratoire d’Informatique, de Robotique et de Microélectronique de Montpellier (LIRMM)

Bio:

Michael Poss is a senior research fellow at the CNRS, in the LIRMM laboratory. He obtained his PhD degree in 2011 at the Université Libre de Bruxelles (2011) under the supervision of Bernard Fortz, Martine Labbé, and François Louveaux. His current research focuses mainly on robust combinatorial optimization. He his the founding managing editor of the Open Journal of Mathematical Optimization.

Abstract

Many discrete optimization problems amount to select a feasible subgraph of least weight. We consider in this work 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 in the chosen subgraph for the worst positions of the vertices in their uncertainty sets. We first prove that these problems are NP-hard even when the feasible subgraphs consist either of all spanning trees or of all s-t paths. In view of this, we propose en 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 two types of polynomial-time approximation algorithms. The first one relies on solving a nominal counterpart of the problem considering pairwise worst-case distances. We study in details the resulting approximation ratio, which depends on the structure of the metric space and of the feasible subgraphs. The second algorithm considers the special case of s-t paths and leads to a fully-polynomial time approximation scheme. Our algorithms are numerically illustrated on a subway network design problem and a facility location problem.