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.

翻译：集合函数可通过多种方式扩展至单位立方体；相关间隙衡量两种自然扩展之间的比率。该量值已被确定为一系列近似算法与机制设计场景中的性能保证。已知单调子模函数的相关间隙至少为$1-1/e$，且该下界对简单拟阵秩函数是紧的。本文开启了对拟阵秩函数相关间隙的细粒度研究。具体而言，我们提出了以拟阵秩和围长为参数的相关间隙改进下界。同时证明对任意拟阵，其加权拟阵秩函数的相关间隙在均匀权重下达到最小。此类改进下界可直接应用于拟阵约束下的子模最大化、机制设计及争用化解方案。