Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An Approximation Framework for Parametric Matroid Interdiction Problems

Nils Hausbrandt, Levin Nemesch, Stefan Ruzika

TL;DR

This work tackles the multi-parametric matroid $\ell$-interdiction problem, where element weights are affine in a parameter vector $\lambda$ drawn from a polytope $\Lambda$. The authors introduce a lifting framework that, given a polynomial-time $\beta$-approximation for fixed $\lambda$, yields a polynomial-time $(1-\varepsilon)\beta$-approximation across the entire $\Lambda$ by decomposing $\Lambda$ into cells via a hyperplane arrangement and solving a common multi-parametric maximization on each cell using a Benson-type algorithm. The approach achieves an FPTAS for multi-parametric partition matroid interdiction and a $(1-\varepsilon)\frac{1}{4}$-approximation for multi-parametric graphic matroid interdiction, and provides the first approximation algorithm for a broad class of multi-parametric optimization problems on arbitrary polytopes. These results enable robust, parametric interdiction strategies in diverse matroid settings with practical implications for network design and optimization under uncertainty.

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 $β$-approximation algorithm for the non-parametric problem, we obtain an approximation algorithm for the multi-parametric problem with an approximation quality arbitrarily close to $β$. 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.

An Approximation Framework for Parametric Matroid Interdiction Problems

TL;DR

This work tackles the multi-parametric matroid -interdiction problem, where element weights are affine in a parameter vector drawn from a polytope . The authors introduce a lifting framework that, given a polynomial-time -approximation for fixed , yields a polynomial-time -approximation across the entire by decomposing into cells via a hyperplane arrangement and solving a common multi-parametric maximization on each cell using a Benson-type algorithm. The approach achieves an FPTAS for multi-parametric partition matroid interdiction and a -approximation for multi-parametric graphic matroid interdiction, and provides the first approximation algorithm for a broad class of multi-parametric optimization problems on arbitrary polytopes. These results enable robust, parametric interdiction strategies in diverse matroid settings with practical implications for network design and optimization under uncertainty.

Abstract

Matroid interdiction problems are well-researched in the field of combinatorial optimization. In the matroid -interdiction problem, an interdiction strategy removes a subset of cardinality 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 -interdiction problem to its multi-parametric variant. Whenever there exists a -approximation algorithm for the non-parametric problem, we obtain an approximation algorithm for the multi-parametric problem with an approximation quality arbitrarily close to . Our method yields an FPTAS for partition matroids and a -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.

Paper Structure

This paper contains 6 sections, 7 theorems, 15 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The approximation framework
  4. Approximation of a Single Cell
  5. Applications
  6. Conclusion

Key Result

Lemma 3.1

On a cell $c\in A(H^=)$, each function $y_F$ is linear and, hence, their upper envelope is convex.

Figures (2)

  • Figure 1: The polyhedron $\mathcal{A}$ after the initialization phase: The initial interdiction strategy $F_0$ was computed by $\mathsf{ORC}_\beta(\lambda_0)$ and the function $w(B_{\lambda_0}^{F_0},\lambda)$ determines a lower bound on $y(\lambda)$ on the interval $c$.
  • Figure 2: The polyhedron $\mathcal{A}$ after the first iteration: The strategy $F$ was computed by $\mathsf{ORC}_\beta(\lambda_{\max})$. Since $(1-\varepsilon)w(B_{\lambda_0}^{F},\lambda_{\max})>w(B_{\lambda_0}^{F_0},\lambda_{\max})$, the function $w(B_{\lambda_0}^{F},\lambda)$ is used to improve the lower bound on $y(\lambda)$.

Theorems & Definitions (18)

  • Definition 2.1: Separating hyperplane (seipp2013adjacency)
  • Definition 2.2: Arrangement
  • Definition 2.4: Set of $\ell$-most vital elements
  • Definition 2.5: Optimal interdiction value function
  • Remark 2.7
  • Definition 2.9
  • Definition 2.10
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • ...and 8 more