Motivated by the growing interest in correlation-robust stochastic optimization, we investigate stochastic selection problems beyond independence. Specifically, we consider the instructive case of pairwise-independent priors and matroid constraints. We obtain essentially-optimal bounds for offline contention resolution and prophet inequalities against the almighty online adversary. The impetus for our work comes from the recent work of \cite{pi-uniform-prophet}, who derived a constant-approximation for the single-choice prophet inequality with pairwise-independent priors. For general matroids, our results are tight and largely negative. For both contention resolution and prophet inequalities, our impossibility results hold for the full linear matroid over a finite field. We explicitly construct pairwise-independent distributions which rule out an $\omega\left(\frac{1}{\rank}\right)$-balanced offline CRS and an $\omega\left(\frac{1}{\log \rank}\right)$-competitive prophet inequality. For both results, we employ a generic approach for constructing pairwise-independent random vectors -- one which unifies and generalizes existing pairwise-independence constructions from the literature on universal hash functions and pseudorandomness. Specifically, our approach is based on our observation that random linear maps turn linear independence into stochastic independence. We then examine the class of matroids which satisfy the so-called partition property -- these include most common matroids encountered in optimization. We obtain positive results for both contention resolution and prophet inequalities with pairwise-independent priors on such matroids, approximately matching the corresponding guarantees for fully independent priors.

翻译：受相关性稳健随机优化日益增长的关注所驱动，我们研究了超越独立性的随机选择问题。具体而言，我们考虑了成对独立先验与拟阵约束这一具有指导意义的案例。针对全能的在线对手，我们获得了离线竞争解决与先知不等式问题的最优界。本研究的动力源自近期文献\cite{pi-uniform-prophet}的工作，该工作推导了成对独立先验下单一选择先知不等式的常数近似比。对于一般拟阵，我们的结果是紧致的且大部分为否定性结论。在竞争解决与先知不等式两方面，我们的不可能性结果均适用于有限域上的全线性拟阵。我们明确构造了成对独立分布，排除了$\omega\left(\frac{1}{\rank}\right)$-平衡离线CRS与$\omega\left(\frac{1}{\log \rank}\right)$-竞争先知不等式的可能性。针对这两个结果，我们采用了构造成对独立随机向量的通用方法——该方法统一并推广了通用哈希函数与伪随机性文献中现有的成对独立性构造。具体而言，该方法基于我们的观察：随机线性映射将线性独立性转化为随机独立性。随后，我们检验了满足所谓划分性质的拟阵类别——这涵盖了优化中遇到的大多数常见拟阵。在此类拟阵上，我们针对成对独立先验获得了竞争解决与先知不等式的正面结果，其近似比与完全独立先验下的相应保证大致匹配。