On the Correlation Gap of Matroids

TL;DR

This paper investigates the correlation gap of matroid rank functions, providing improved bounds based on matroid parameters, with implications for optimization, mechanism design, and contention resolution.

Contribution

It offers a refined analysis of the correlation gap for matroid rank functions, including bounds depending on rank and girth, and shows minimization under uniform weights.

Findings

Improved lower bounds on the correlation gap based on matroid parameters.

Correlation gap minimized under uniform weights for weighted matroids.

Applications to submodular maximization and mechanism design.

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.