Randomized Max-Vertex-Cover Interdiction with Matroid Constraints

TL;DR

This paper introduces an approximation framework for bilevel network interdiction problems, specifically addressing the NP-hard Randomized Max-Vertex-Cover Interdiction with matroid constraints, and provides a polynomial-time 8/3-approximation algorithm.

Contribution

It develops a novel approximation framework for bilevel interdiction problems and achieves the first polynomial-time 8/3-approximation for RMVCI under matroid constraints.

Findings

Linear relaxation of follower's problem has a tight integrality gap of 4/3.

A polynomial-time 2-approximation algorithm is designed for the relaxed bilevel problem.

The framework yields an 8/3-approximation algorithm for the original RMVCI problem.

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.