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：受相关性鲁棒随机优化日益增长的兴趣驱动，我们研究了超越独立假设的随机选择问题。具体而言，我们考虑了具有指导意义的成对独立先验和拟阵约束情形。针对离线争用解决和先知不等式问题，我们获得了对抗全能在线对手的本质上最优的界。本工作的动力来自近期文献\cite{pi-uniform-prophet}，该文献针对具有成对独立先验的单选择先知不等式导出了常数近似比。对于一般拟阵，我们的结果是紧的且基本为否定性结论。对于争用解决和先知不等式，我们的不可能性结果在有限域上的全线性拟阵上成立。我们显式构造了成对独立分布，排除了\(\omega\left(\frac{1}{\rank}\right)\)平衡离线CRS和\(\omega\left(\frac{1}{\log \rank}\right)\)竞争比先知不等式的存在性。对于这两个结果，我们采用了一种构造成对独立随机向量的通用方法——该方法统一并推广了现有文献中关于通用哈希函数和伪随机性的成对独立构造。具体而言，我们的方法基于如下观察：随机线性映射将线性独立性转化为随机独立性。接着，我们考察了满足所谓划分性质的一类拟阵——这包括优化中遇到的大多数常见拟阵。我们获得了此类拟阵上具有成对独立先验的争用解决和先知不等式的正向结果，其近似比与完全独立先验下的相应保证基本匹配。