Optimal Contention Resolution from Samples
Contention resolution schemes convert random sets which are ex-ante feasible for some constraint into sets which are feasible ex post, and their online variants have broad applications in Bayesian and stochastic optimization, including prophet inequalities and stochastic probing. In many such applications, however, full distributional knowledge is an unattainable requirement. This motivates the study of contention resolution schemes which employ a limited number of samples from the underlying distributions. We study sample-based contention resolution for $k$-uniform matroids, for both uniformly random and adversarial arrival orders, and compare their balance guarantees to the best possible with full distributional knowledge. For a single-item constraint, we show that one sample per element suffices for an optimal $(1-1/e)$-balanced ROCRS. For adversarial arrivals, however, we prove that no single-sample OCRS can be $1/2$-balanced, even for known arrival orders, separating contention resolution from prophet inequalities in the single-sample setting. Perhaps surprisingly, we show that two samples are necessary and sufficient for the optimal $1/2$ guarantee. We also present a single-sample $0.48$-balanced scheme. For general $k$, we give a simple single-sample ROCRS that achieves an optimal balance ratio of $1-e^{-k}k^k/k!$. In adversarial order, we obtain a single-sample $(1 - O(1/\sqrt{k}))$-balanced scheme, matching the optimal asymptotic dependence on $k$. Finally, we complete the picture by characterizing the minimum number of samples needed to match the optimal balance ratio achievable with full distributional knowledge. For random arrivals, one sample suffices for $k = 1,2,3,4$, and zero samples suffice for $k \geq 5$. For an adversarial arrival order, even known in advance, and any $k \geq 2$, we show that no constant number of samples suffices to achieve optimal balance.