The Parametric Matroid $\ell$-Interdiction Problem

TL;DR

This paper introduces the parametric matroid $ ext{ extlbrackdbl} ext{ell} ext{ extrbrackdbl}$-interdiction problem, analyzing how to identify the most critical elements in a matroid as a parameter varies, with polynomial-time algorithms under certain conditions.

Contribution

It defines a new parametric interdiction problem for matroids and provides polynomial-time algorithms for solving it when independence testing is efficient.

Findings

The set of most vital elements changes polynomially with the parameter.

Polynomial algorithms are developed for the interdiction problem.

The problem's complexity depends on the efficiency of independence testing.

Abstract

In this article, we introduce the parametric matroid $\ell$-interdiction problem, where $\ell\in\mathbb{N}_{>0}$ is a fixed number of elements allowed to be interdicted. Each element of the matroid's ground set is assigned a weight that depends linearly on a real parameter from a given interval. The goal is to compute, for each possible parameter value, a set of $\ell$-most vital elements with corresponding objective value the deletion of which causes a maximum increase of the weight of a minimal basis. We show that such a set, which of course depends on the parameter, can only change polynomially often if the parameter varies. We develop several exact algorithms to solve the problem that have polynomial running times if an independence test can be performed in polynomial time.