An Approximation Framework for Parametric Matroid Interdiction Problems

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.