Interdicting Structured Combinatorial Optimization Problems with {0,1}-Objectives

TL;DR

This paper introduces a general method for approximating interdiction problems in combinatorial optimization, providing pseudo-approximations that balance solution quality and interdiction budget overrun, applicable to various problem types.

Contribution

The authors propose a versatile pseudo-approximation framework for interdiction problems, capable of handling submodular costs and improving results with problem-specific structural insights.

Findings

Provides a pseudo-approximation algorithm for interdiction problems.

Handles submodular interdiction costs in maximum weight independent set.

Develops a PTAS for interdiction of b-stable sets in bipartite graphs.

Abstract

Interdiction problems ask about the worst-case impact of a limited change to an underlying optimization problem. They are a natural way to measure the robustness of a system, or to identify its weakest spots. Interdiction problems have been studied for a wide variety of classical combinatorial optimization problems, including maximum $s$-$t$ flows, shortest $s$-$t$ paths, maximum weight matchings, minimum spanning trees, maximum stable sets, and graph connectivity. Most interdiction problems are NP-hard, and furthermore, even designing efficient approximation algorithms that allow for estimating the order of magnitude of a worst-case impact, has turned out to be very difficult. Not very surprisingly, the few known approximation algorithms are heavily tailored for specific problems. Inspired by an approach of Burch et al. (2003), we suggest a general method to obtain pseudoapproximations for many interdiction problems. More precisely, for any $\alpha>0$, our algorithm will return either a $(1+\alpha)$-approximation, or a solution that may overrun the interdiction budget by a factor of at most $1+\alpha^{-1}$ but is also at least as good as the optimal solution that respects the budget. Furthermore, our approach can handle submodular interdiction costs when the underlying problem is to find a maximum weight independent set in a matroid, as for example the maximum weight forest problem. The approach can sometimes be refined by exploiting additional structural properties of the underlying optimization problem to obtain stronger results. We demonstrate this by presenting a PTAS for interdicting $b$-stable sets in bipartite graphs.