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：集合函数可以以多种方式扩展到单位立方体；相关性间隙测量了两种自然扩展之间的比率。该量已被确定为一系列近似算法和机制设计设置中的性能保证。已知单调次模函数的相关性间隙至少为$1-1/e$，并且这对于简单拟阵秩函数是紧的。我们启动了对拟阵秩函数相关性间隙的精细研究。特别地，我们提出了以拟阵的秩和围长参数化的相关性间隙改进下界。我们还证明，对于任何拟阵，其加权拟阵秩函数的相关性间隙在均匀权重下达到最小值。这些改进的下界在拟阵约束下的次模最大化、机制设计和冲突解决方案中具有直接应用。