On the correlation gap of matroids

• 成果类型: Article
• 署名作者: Husic, Edin; Koh, Zhuan Khye; Loho, Georg; Vegh, Laszlo A.
• 署名单位: Universita della Svizzera Italiana; University of Twente; University of London; London School Economics & Political Science
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-024-02116-w
• 发表日期: 2025
• 页码: 407-456
• 关键词: submodular function maximization; function subject; algorithm

摘要

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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1-1/e$$\end{document}, 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.