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 contention resolution and prophet inequalities. The impetus for our work comes from the recent work of Caragiannis et al., 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(1/Rank)-balanced offline CRS and an omega(1/log Rank)-competitive prophet inequality against the (usual) oblivious adversary. 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 online contention resolution and prophet inequalities with pairwise-independent priors on such matroids, approximately matching the corresponding guarantees for fully independent priors. These algorithmic results hold against the almighty adversary for both problems.

翻译：受相关性鲁棒随机优化领域日益增长的兴趣驱动，我们研究了超越独立性假设的随机选择问题。具体而言，我们考虑成对独立先验与拟阵约束这一具有启发性的情形。对于冲突消解和先知不等式，我们获得了本质上最优的界。本研究的直接动因来自Caragiannis等人的最新工作，他们针对成对独立先验下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在很大程度上是消极的：在冲突消解与先知不等式两个问题中，不可行性结论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，其排除了在离线CRS中实现ω(1/Rank)均衡比、以及在（通常的）对抗性对手下实现ω(1/log Rank)竞争比的先知不等式。为获得这两个结论，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与伪随机性的成对独立性构造。具体而言，我们的方法基于如下观察：随机线性映射能够将线性独立性转化为随机独立性。随后，我们研究满足所谓划分性质的一类拟阵——优化问题中常见的大多数拟阵均属此类。对于此类拟阵上的成对独立先验，我们在在线冲突消解与先知不等式两个问题上均获得了正面结果，其近似程度与完全独立先验下的保证相匹配。这些算法性结论在面对全能型对手时仍然成立。