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On the Correlation Gap of Matroids

Edin Husić, Zhuan Khye Koh, Georg Loho, László A. Végh

TL;DR

We address the correlation gap for matroid rank functions, refining the universal $1-1/e$ bound by deriving a rank- and girth-dependent lower bound and proving that uniform weights minimize the gap. The analysis combines a Poisson-clock technique with a two-stage decomposition into a uniform matroid component and a residual, showing the minimizer lies in the independent-set polytope and yields integral mass. These results yield tighter guarantees for submodular maximization under matroid constraints, as well as for sequential posted-price mechanisms and contention-resolution schemes, connecting algorithmic performance to fine-grained matroid structure. The findings illuminate how matroid parameters influence approximation ratios and mechanism design guarantees, with implications for CR schemes and concave-coverage problems, and suggest directions for tighter bounds for special matroid classes.

Abstract

A set function can be extended to the unit cube in various ways; the correlation gap measures the ratio between two natural extensions. This quantity has been identified as the performance guarantee in a range of approximation algorithms and mechanism design settings. It is known that the correlation gap of a monotone submodular function is at least $1-1/e$, and this is tight for simple matroid rank functions. We initiate a fine-grained study of the correlation gap of matroid rank functions. In particular, we present an improved lower bound on the correlation gap as parametrized by the rank and girth of the matroid. We also show that for any matroid, the correlation gap of its weighted matroid rank function is minimized under uniform weights. Such improved lower bounds have direct applications for submodular maximization under matroid constraints, mechanism design, and contention resolution schemes.

On the Correlation Gap of Matroids

TL;DR

We address the correlation gap for matroid rank functions, refining the universal bound by deriving a rank- and girth-dependent lower bound and proving that uniform weights minimize the gap. The analysis combines a Poisson-clock technique with a two-stage decomposition into a uniform matroid component and a residual, showing the minimizer lies in the independent-set polytope and yields integral mass. These results yield tighter guarantees for submodular maximization under matroid constraints, as well as for sequential posted-price mechanisms and contention-resolution schemes, connecting algorithmic performance to fine-grained matroid structure. The findings illuminate how matroid parameters influence approximation ratios and mechanism design guarantees, with implications for CR schemes and concave-coverage problems, and suggest directions for tighter bounds for special matroid classes.

Abstract

A set function can be extended to the unit cube in various ways; the correlation gap measures the ratio between two natural extensions. This quantity has been identified as the performance guarantee in a range of approximation algorithms and mechanism design settings. It is known that the correlation gap of a monotone submodular function is at least , and this is tight for simple matroid rank functions. We initiate a fine-grained study of the correlation gap of matroid rank functions. In particular, we present an improved lower bound on the correlation gap as parametrized by the rank and girth of the matroid. We also show that for any matroid, the correlation gap of its weighted matroid rank function is minimized under uniform weights. Such improved lower bounds have direct applications for submodular maximization under matroid constraints, mechanism design, and contention resolution schemes.
Paper Structure (36 sections, 37 theorems, 172 equations, 1 figure)

This paper contains 36 sections, 37 theorems, 172 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Role of Correlation Gap
  3. Monotone Submodular Maximization
  4. Concave Coverage Problems
  5. Sequential Posted Price Mechanisms
  6. Contention Resolution Schemes
  7. Our Results
  8. Our Techniques
  9. Weighted Rank Functions
  10. Uniform Matroids
  11. Poisson Clock Analysis
  12. A Two Stage Approach
  13. Further Related Work
  14. Paper Organization
  15. Preliminaries
  16. ...and 21 more sections

Key Result

Proposition 1.1

Let $f_1,f_2,\ldots,f_m \colon 2^E\to{{\mathbb{R}}}$ be monotone $M^\natural$-concave functions, and let $f=\sum_{i=1}^m f_i$. Then, a $\min_{i=1}^m {{\mathcal{CG}}}(f_i)$-approximation algorithm for prob:submod-max can be obtained in time polynomial in $|E|$, $m$ and $\mu(f)$.

Figures (1)

  • Figure 1: Our correlation gap bound as a function of the rank $\rho$, and as a function of the girth $\gamma$ separately.

Theorems & Definitions (94)

  • Proposition 1.1
  • Theorem 1.2: yan2011mechanism
  • Theorem 1.3
  • Proposition 1.4
  • Theorem 1.5
  • Theorem 2.1: Edmonds, Schrijverbook
  • Theorem 2.2: Edmonds, Schrijverbook
  • Proposition 2.3: Murotabook
  • Proposition 2.4: DressWenzel:1992
  • Proposition 2.5
  • ...and 84 more