arXiv:2611.16880 · v1

Tight Bounds for Downward-Closed Prophet and Secretary Problems

向下封闭预言家与秘书问题的紧确界

首次提交 2026-11-12 22:44

版本更新 2026-11-12 22:44

雷达观测 2026-11-13 14:15

摘要

We study online selection problems subject to arbitrary downward-closed feasibility constraints, under different arrival orders, with different distributional assumptions. We establish the following bounds: (1) $Θ(\log^2 n / \log \log n)$ for the standard (adversarial-order) prophet problem, (2) $Ω(\log n)$ for the (Bayesian, random-order) prophet secretary problem, and by extension, for the standard (prior-free, random-order) secretary problem, and (3) $O(\log n / \log \log n)$ for the free-order prophet problem. Together with complementary bounds established in prior work, these results pin down the asymptotically tight competitive ratios for all these variants of prophet and secretary problems subject to downward-closed constraints. Our results refine and reinforce the big-decisions-first principle by Rubinstein and Singla [2026], establishing a superconstant separation between the random-order and free-order models.