Branch-and-price strikes back for the k-vertex cut problem

TL;DR

This paper introduces a new branch-and-price algorithm for the k-vertex cut problem, improving solution efficiency and solving previously unsolved instances through a strengthened ILP formulation and tailored algorithmic techniques.

Contribution

It develops a unified, stronger ILP model and a novel branch-and-price algorithm with specialized components for the k-VCP, outperforming existing methods.

Findings

Outperforms state-of-the-art methods on 608 benchmark instances.

Solves 73 previously unsolved instances within one hour.

The new model dominates all previous ILP formulations in relaxation quality.

Abstract

Given an undirected graph, the k-vertex cut problem (k-VCP) asks for a minimum-cost set of vertices whose removal yields at least k connected components in the resulting graph. The k-VCP is an important problem in network optimization, with applications in infrastructure protection and epidemic containment. We present a new extended integer linear programming (ILP) formulation that unifies and strengthens existing models and serves as the foundation for a new branch-and-price algorithm for the k-VCP. An in-depth theoretical study enables us to devise algorithmic components such as tailored branching rules that preserve the structure of the pricing problems, as well as valid inequalities and symmetry-handling techniques. We also show that our new model dominates all previous ILP formulations of the k-VCP in terms of their linear relaxations, which theoretically justifies the computational effectiveness of our approach. Extensive computational experiments against state-of-the-art methods demonstrate substantially improved performance, both in terms of instances solved to proven optimality and running times. On the full benchmark of 608 instances, our algorithm consistently outperforms all competitors and is able to solve 73 previously unsolved instances within the time limit of one hour.