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

TL;DR

The paper tackles the NP-hard $k$-vertex cut problem by introducing a new extended ILP, $ILP_E2$, that combines the natural vertex-cut variables with a continuous relaxation of the cluster-selection variables from prior exponential-size models, yielding a strictly stronger LP relaxation than the existing $ILP_N$ and $ILP_E1$. Building on this formulation, the authors develop a branch-and-price algorithm, $BP^*$, featuring a column-generation pricing problem solved via min-cut networks, a tailored branching strategy aligned with pricing, and symmetry-handling techniques to prune equivalent solutions. Computational experiments on 304 unweighted and 304 weighted benchmark instances show that $BP^*$ solves substantially more instances and does so faster than state-of-the-art methods, solving 73 previously open instances within a one-hour limit, and delivering tighter LP bounds across the board. The work demonstrates that a carefully crafted extended formulation coupled with branch-and-price can surpass existing approaches for graph-partitioning problems, and suggests broader applicability to related critical-vertex problems and higher-order vertex-cut variants.

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.

Figures (5)

• Figure 1: Optimal $k$-vertex cuts (gray vertices with dashed incident edges) for Zachary’s karate club network with unit vertex costs, shown for $k \in \{3,5,10\}$. The colored vertices represent the connected components remaining in the graph.
• Figure 2: Illustration of the auxiliary network for solving the pricing problem. The capacities in the network are given next to the arcs and we assume $\sigma = 3$.
• Figure 3: Performance profiles comparing BP$^\star$ with its three variant and BP$^\dagger$ on the benchmark set of unweighted instances, with a time limit of one hour.
• Figure 4: Cumulative percentage of instances solved to optimality over time by each algorithm for unweighted (left) and weighted (right) instances, with a time limit of one hour.
• Figure 5: Performance profiles of algorithms COMP, HYB and BP$^\star$ on the full set of unweighted and weighted benchmark instances, with a time limit of one hour.

Theorems & Definitions (27)

• Definition 1: $k$-vertex cut problem
• Theorem 1
• Lemma 1
• proof
• Theorem 2
• proof
• Corollary 1
• proof
• Theorem 3
• Lemma 2