Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Weighted-to-Unweighted Reduction for Matroid Intersection

Aditi Dudeja, Mara Grilnberger

Abstract

Given two matroids $\mathcal{M}_1$ and $\mathcal{M}_2$ over the same ground set, the matroid intersection problem is to find the maximum cardinality common independent set. In the weighted version of the problem, the goal is to find a maximum weight common independent set. It has been a matter of interest to find efficient approximation algorithms for this problem in various settings. In many of these models, there is a gap between the best known results for the unweighted and weighted versions. In this work, we address the question of closing this gap. Our main result is a reduction which converts any $α$-approximate unweighted matroid intersection algorithm into an $α(1-\varepsilon)$-approximate weighted matroid intersection algorithm, while increasing the runtime of the algorithm by a $\log W$ factor, where $W$ is the aspect ratio. Our framework is versatile and translates to settings such as streaming and one-way communication complexity where matroid intersection is well-studied. As a by-product of our techniques, we derive new results for weighted matroid intersection in these models.

A Weighted-to-Unweighted Reduction for Matroid Intersection

Abstract

Given two matroids and over the same ground set, the matroid intersection problem is to find the maximum cardinality common independent set. In the weighted version of the problem, the goal is to find a maximum weight common independent set. It has been a matter of interest to find efficient approximation algorithms for this problem in various settings. In many of these models, there is a gap between the best known results for the unweighted and weighted versions. In this work, we address the question of closing this gap. Our main result is a reduction which converts any -approximate unweighted matroid intersection algorithm into an -approximate weighted matroid intersection algorithm, while increasing the runtime of the algorithm by a factor, where is the aspect ratio. Our framework is versatile and translates to settings such as streaming and one-way communication complexity where matroid intersection is well-studied. As a by-product of our techniques, we derive new results for weighted matroid intersection in these models.
Paper Structure (39 sections, 41 theorems, 30 equations, 1 figure, 1 table, 3 algorithms)

This paper contains 39 sections, 41 theorems, 30 equations, 1 figure, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Matroid Intersection.
  3. Accessing a Matroid.
  4. Related Work and Dependence on $\varepsilon$.
  5. Overview of Techniques
  6. Notation.
  7. Linear Programming for Matroid Intersection.
  8. Our Framework
  9. Aspect-Ratio Reduction.
  10. Weighted-to-Unweighted Reduction.
  11. Refolding.
  12. Preliminaries
  13. Roadmap.
  14. Ingredients for Our Framework
  15. Matroid Intersection Unfolding
  16. ...and 24 more sections

Key Result

Lemma 2.1

We say that a sequence of sets $S_1,S_2,\cdots, S_l$ form a chain if $S_1\subsetneq S_2\subsetneq S_3\subsetneq\cdots\subsetneq S_l$. There exists optimal solutions $y,z$ for Linear Program eqn:dualunweighted such that $\textup{supp}(y)$ forms a chain and $\textup{supp}(z)$ forms a chain. This also

Figures (1)

  • Figure 1: An example of matroid unfolding for two graphic matroids $\mathop{\mathrm{\mathcal{M}}}\nolimits'_1$, $\mathop{\mathrm{\mathcal{M}}}\nolimits'_2$. In a graphic matroid, the elements are given by the edges of a graph and the independent sets are the forests. The labels of edges in $\mathop{\mathrm{\mathcal{M}}}\nolimits'_1,\mathop{\mathrm{\mathcal{M}}}\nolimits'_2$ contain the name of the edge as well as the associated weight. The blue edges show a maximum weight common indpendent set in $\mathop{\mathrm{\mathcal{M}}}\nolimits'_1,\mathop{\mathrm{\mathcal{M}}}\nolimits'_2$ and a maximum cardinality common independent set with the same value in the unfolded matroids $\mathop{\mathrm{\mathcal{M}}}\nolimits_1$, $\mathop{\mathrm{\mathcal{M}}}\nolimits_2$. The cycles of the new graphic matroids correspond to cycles in the original instance.

Theorems & Definitions (87)

  • Lemma 2.1: Edmonds03
  • Definition 2.2: Span
  • Definition 2.3: Circuits
  • Definition 3.1: Unfolded Matroid Intersection
  • Lemma 3.2
  • proof
  • Claim 1: Independence Queries
  • Claim 2: Rank Queries
  • Theorem 3.3
  • Lemma 3.4
  • ...and 77 more