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Multi-parametric matroids -- Applications to interdiction and weight set decomposition

Nils Hausbrandt, Stefan Ruzika

TL;DR

This work introduces the multi-parametric matroid problem, where element weights depend linearly on $p$ parameters and seeks a minimum-weight basis for every parameter vector $oldsymbol{\lambda}\

Abstract

In this article, we investigate the multi-parametric matroid problem. The weights of the elements of the matroid's ground set depend linearly on an arbitrary but fixed number of parameters, each of which is taken from a real interval. The goal is to compute a minimum weight basis for each possible combination of the parameters. For this problem, we propose an algorithm that requires a polynomial number of independence tests and discuss two useful applications. First, the algorithm can be applied to solve a multi-parametric version of a special matroid interdiction problem, and second, it can be utilized to compute the weight set decomposition of the multi-objective (graphic) matroid problem. For the latter, we asymptotically improve the current state-of-the-art algorithm by a factor that is almost proportional to the number of edges of the graphic matroid.

Multi-parametric matroids -- Applications to interdiction and weight set decomposition

TL;DR

This work introduces the multi-parametric matroid problem, where element weights depend linearly on parameters and seeks a minimum-weight basis for every parameter vector $oldsymbol{\lambda}\

Abstract

In this article, we investigate the multi-parametric matroid problem. The weights of the elements of the matroid's ground set depend linearly on an arbitrary but fixed number of parameters, each of which is taken from a real interval. The goal is to compute a minimum weight basis for each possible combination of the parameters. For this problem, we propose an algorithm that requires a polynomial number of independence tests and discuss two useful applications. First, the algorithm can be applied to solve a multi-parametric version of a special matroid interdiction problem, and second, it can be utilized to compute the weight set decomposition of the multi-objective (graphic) matroid problem. For the latter, we asymptotically improve the current state-of-the-art algorithm by a factor that is almost proportional to the number of edges of the graphic matroid.

Paper Structure

This paper contains 6 sections, 13 theorems, 17 equations, 2 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Algorithms
  4. Application 1: Weight set decomposition for multi-objective matroids
  5. Application 2: Multi-parametric matroid one-interdiction
  6. Conclusion

Key Result

Lemma 2.5

The optimal value function $w$ is piecewise linear, continuous and concave.

Figures (2)

  • Figure 1: An instance of the 2-parametric minimum spanning tree problem.
  • Figure 3: The arrangement of the separating hyperplanes $A(H^=)$ obtained by intersecting the 2-parametric weights of the edges from the graph in \ref{['fig:ex_graph_parametric_p_1']}. Each cell is assigned its unique minimum spanning tree. The minimum spanning trees of two neighboring cells differ by a maximum of two edges. If two cells have the same minimum spanning tree, they are colored in the same shade of gray.

Theorems & Definitions (32)

  • Definition 2.2: Optimal value function
  • Definition 2.3: Separating hyperplane (seipp2013adjacency)
  • Lemma 2.5
  • proof
  • Example 2.7
  • Definition 2.8: edelsbrunner1986constructing
  • Theorem 2.9: seipp2013adjacency
  • Theorem 2.10: edelsbrunner1986constructing
  • Definition 2.11: Incidence graph (grunbaum1967convex)
  • Theorem 2.12: grunbaum1967convex
  • ...and 22 more