2021/09/01 by Marin Bougeret, Bougeret, Marin, Jérémy Omer +3
Business, Management and Accounting · Decision Sciences · Engineering · #68Q27 #68W25 #Data Structures and Algorithms (cs.DS) #F.2.2 #FOS: Computer and information sciences #Facility Location and Emergency Management #Risk and Portfolio Optimization #Vehicle Routing Optimization Methods
paper · doi:10.48550/arxiv.2109.00389
openalex publication_date 2021/09/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.