受限独立性下依赖维度相关间隙界
Dimension Dependent Correlation Gap Bounds under Restricted Independence
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中文总结 AI 辅助
该研究解决了成对独立相关间隙的$n=4$界及最坏情形紧界问题，证明$n=4$时$4/3$界普遍成立且紧，最坏情形下成对独立相关间隙渐近达$e/(e-1)$，并推广到$t$阶独立（$t\ge2$）。
中文摘要 AI 辅助
成对独立相关间隙是集合函数在任意依赖下的最大期望值与成对独立下的最大期望值之比，用于衡量该独立性限制带来的损失。对于单调次模函数，在相互独立条件下，该间隙普遍以$e/(e-1)$为界；在成对独立条件下，针对包括$n=3$在内的若干特殊情形，已建立更紧的$4/3$上界，并被猜测普遍成立。近期一项AI辅助的反例否定了$n=5$时的该猜测，使得$n=4$时的界以及最坏情形下的紧界的有效性成为悬而未决的问题。我们解决了这两个问题。首先，对于$n=4$，我们结合理论分析与计算验证，通过AI辅助证明确定$4/3$界普遍成立且是紧的。该证明结合了最优分子顶点的结构表征、置换对称性、锥证书系统、伯恩斯坦多项式表示、递归单纯形细分，以及对2745个伯恩斯坦系数系统的验证。其次，我们通过构造一个边际概率相同的实例，将基集划分为$m$个块，采用单调次模的并覆盖函数，证明最坏情形下的成对独立相关间隙渐近达到$e/(e-1)$，且块数随基集大小亚线性增长。该结果通过构造成对独立线性规划的缩放渐近约化对偶的可行解得到，且由于$t$阶独立（$t\ge2$）蕴含成对独立，该结果可直接推广到$t$阶独立随机元素。因此，尽管成对独立是$t$阶独立层级中限制性最弱的独立性形式，但在最坏情形下，其限制性可与相互独立相当。
英文摘要
The pairwise independent correlation gap is the ratio of the maximum expected value of a set function under arbitrary dependence to that under pairwise independence, measuring the loss from this independence restriction. Under mutual independence, this gap is universally bounded by $e/(e-1)$ for monotone submodular functions. With pairwise independence, a tighter $4/3$ upper bound was established for several special cases, including $n=3$, and conjectured to hold universally. A recent AI-assisted counterexample disproved this conjecture for $n=5$, leaving the validity of the $n=4$ bound and the tight worst case bound open. We resolve both questions. First, for $n=4$, we establish that the $4/3$ bound holds universally and is tight using an AI-assisted proof combining theoretical analysis and computational verification. The proof combines a structural characterization of optimal numerator vertices, permutation symmetry, cone certificate systems, Bernstein polynomial representations, recursive simplex subdivision, and verification of $2,745$ Bernstein coefficient systems. Second, we show that the worst case pairwise independent correlation gap attains $e/(e-1)$ asymptotically by constructing an instance with identical marginal probabilities and a monotone submodular union coverage function on a ground set partitioned into $m$ blocks. The number of blocks grows sublinearly with the ground set size. The result follows by constructing a feasible solution to a scaled asymptotic reduced dual of the pairwise independent linear program and immediately extends to $t$-wise independent random elements ($t\ge2$), since $t$-wise independence implies pairwise independence. Thus, pairwise independence, despite being the least restrictive form of independence in the $t$-wise independence hierarchy, can be as restrictive as mutual independence in the worst case.
发表机构
- Indian Institute of Management Bangalore（印度管理学院班加罗尔分校）
机构由 AI 辅助整理，请以论文原文为准。
