An Approximation Framework for Parametric Matroid Interdiction Problems

TL;DR

This paper introduces a novel framework that extends approximation algorithms for matroid interdiction problems to their multi-parametric versions, enabling efficient solutions for complex combinatorial optimization scenarios.

Contribution

It develops the first general approach to lift approximation algorithms from non-parametric to multi-parametric matroid interdiction problems, including an FPTAS for partition matroids.

Findings

Provides an FPTAS for partition matroids.

Achieves a (1-ε)/4-approximation for graphic matroids.

Develops the first approximation algorithm for multi-parametric optimization with parameters in an arbitrary polytope.

Abstract

Matroid interdiction problems are well-researched in the field of combinatorial optimization. In the matroid $\ell$-interdiction problem, an interdiction strategy removes a subset of cardinality $\ell$ from the matroid's ground set. The goal is to maximize the weight of a remaining optimal basis. We examine the multi-parametric generalization of this problem, where every weight is given by a linear function depending on a parameter vector. For every parameter value, we are interested in an optimal interdiction strategy and the weight of an optimally interdicted basis. We develop the first framework for lifting approximation algorithms for the non-parametric matroid $\ell$-interdiction problem to its multi-parametric variant. Whenever there exists a $\beta$-approximation algorithm for the non-parametric problem, we obtain an approximation algorithm for the multi-parametric problem with an approximation quality arbitrarily close to $\beta$. Our method yields an FPTAS for partition matroids and a $(1-\varepsilon)\frac{1}{4}$-approximation for graphic matroids. As part of the construction, we develop the first approximation algorithm for a conventional multi-parametric optimization problem in which the parameter vector varies in an arbitrary polytope.