Randomized Max-Vertex-Cover Interdiction with Matroid Constraints

2026-05-11

AI Summary

This study addresses the Robust Matroid Vertex Coverage Interdiction (RMVCI) problem, a bilevel optimization task under matroid constraints where a leader randomly selects vertices to protect in order to minimize the maximum expected edge-weight gain achievable by an attacker, formulated as a zero-sum Stackelberg game. Recognizing the problem’s NP-hardness, the authors propose a general bilevel interdiction approximation framework and present the first polynomial-time 8/3-approximation algorithm. They also establish that the integrality gap of the follower’s linear programming relaxation is tightly bounded by 4/3. By integrating integer linear programming modeling, LP relaxation, probabilistic distribution transformation, and bilevel optimization approximation techniques, the approach substantially enhances the tractability of this class of problems.

Abstract

We study a new bilevel optimization problem, termed the Randomized Max-Vertex-Cover Interdiction (RMVCI) problem under matroid constraints, which can be modeled as a zero-sum Stackelberg game on a network between a leader and a follower. The leader randomly selects a subset of vertices to protect, subject to a matroid constraint, while the follower-after inferring the leader's protection probability distribution-chooses a subset of vertices (also matroid-constrained) to attack, aiming to maximize the expected total weight of edges incident to the set of vertices that are both attacked and unprotected. The leader's objective is to determine an optimal randomized interdiction strategy that minimizes the follower's expected payoff. Since the follower's response problem is NP-hard, the resulting bilevel program is computationally challenging. We develop a conceptual approximation framework for tackling general bilevel interdiction problems. For the RMVCI problem under matroid constraints, we first formulate the follower's problem as an integer linear program and show that its linear relaxation admits a tight integrality gap of $\tfrac{4}{3}$. Within the approximation framework, we replace the follower's problem by its LP relaxation, and then study the resulting bilevel program. By shifting from distributions over sets to distributions over vertices and applying our approximation framework, we manage to design a polynomial-time 2-approximation algorithm for this relaxed bilevel problem. Combining these ingredients within our framework yields a polynomial-time $\tfrac{8}{3}$-approximation algorithm for RMVCI under matroid constraints.

Problem

• bilevel optimization
• vertex cover interdiction
• matroid constraints
• Stackelberg game
• randomized strategy

Innovation

• bilevel optimization
• matroid constraints
• approximation algorithm
• Stackelberg game
• vertex cover interdiction