On the correlation gap of matroids

TL;DR

This paper studies the correlation gap of matroid rank functions and provides improved lower bounds based on matroid properties.

Contribution

The paper introduces a fine-grained analysis of matroid rank functions' correlation gaps with improved lower bounds.

Findings

An improved lower bound on the correlation gap is presented based on matroid rank and girth.

The correlation gap of a matroid's weighted rank function is minimized under uniform weights.

The results have implications for 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 \documentclass[12pt]{minimal}
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\begin{document}$$1-1/e$$\end{document}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 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.

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