Limitations of Stochastic Selection Problems with Pairwise Independent Priors

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.