Contribution to Blocker and Interdiction optimization problems in networks
Sébastien Martin
TL;DR
This work advances blocker and interdiction concepts in networks by presenting polyhedral analyses and exact-ILP approaches for classic combinatorial problems, including bipartite complete matching, vertex k-cut, and flow/clique blockers. It develops polynomial-time formulations for the bipartite blocker variant, while establishing NP-hardness for several multi-set and flow-interdiction variants, and introduces Branch-and-Cut and Branch-and-Price algorithms tailored to these problems. A unifying theme is the use of strong valid inequalities and separation/pricing routines to tighten relaxations, enabling large-scale exact computations on problems with up to several hundred vertices. The results demonstrate significant algorithmic gains and provide practical insights for resilience, monitoring, and partitioning in networks, with broad applicability to telecommunication and scheduling contexts.
Abstract
This manuscript describes the notions of blocker and interdiction applied to well-known optimization problems. The main interest of these two concepts is the capability to analyze the existence of a combinatorial structure after some modifications. We focus on graph modification, like removing vertices or links in a network. In the interdiction version, we have a budget for modification to reduce as much as possible the size of a given combinatorial structure. Whereas, for the blocker version, we minimize the number of modifications such that the network does not contain a given combinatorial structure. Blocker and interdiction problems have some similarities and can be applied to well-known optimization problems. We consider matching, connectivity, shortest path, max flow, and clique problems. For these problems, we analyze either the blocker version or the interdiction one. Applying the concept of blocker or interdiction to well-known optimization problems can change their complexities. Some optimization problems become harder when one of these two notions is applied. For this reason, we propose some complexity analysis to show when an optimization problem, or the associated decision problem, becomes harder. Another fundamental aspect developed in the manuscript is the use of exact methods to tackle these optimization problems. The main way to solve these problems is to use integer linear programming to model them. An interesting aspect of integer linear programming is the possibility to analyze theoretically the strength of these models, using cutting planes. For most of the problems studied in this manuscript, a polyhedral analysis is performed to prove the strength of inequalities or describe new families of inequalities. The exact algorithms proposed are based on Branch-and-Cut or Branch-and-Price algorithm, where dedicated separation and pricing algorithms are proposed.