论文标题
通过预期竞争比率的先知不平等
Prophet Inequalities via the Expected Competitive Ratio
论文作者
论文摘要
我们考虑在向下关闭的约束下的先知不平等。在这个问题中,决策者在到达元素上做出了立即和不可撤销的选择,但要受到限制。传统上,性能与预期的离线最佳距离进行了比较,称为\ textit {期望值}(ROE)。但是,ROE具有局限性,因为它仅保证了与最佳效果相比的平均性能,并且与实现的前柱最佳值相比可能表现不佳。我们研究了另一种性能度量,即\ textit {预期比率}(eor),即算法和先知价值之间对比率的期望。 EOR提供了强大的保证,例如,恒定的EOR意味着以恒定的概率达到离线最佳的恒定部分。对于单选问题的特殊情况,EOR与选择最大值的概率的概念相吻合。但是,EOR自然会概括为组合约束选择最大值的概率,这是本文的主要重点。具体来说,我们建立了两个减少:对于每个约束,ROE和EOR最多都是恒定的因素。此外,我们表明EOR比ROE更强,因为对于每个实例(约束和分布),ROE至少是EOR的恒定部分,但反之亦然。这两种减少都意味着大量EOR导致多种环境已知结果。
We consider prophet inequalities under downward-closed constraints. In this problem, a decision-maker makes immediate and irrevocable choices on arriving elements, subject to constraints. Traditionally, performance is compared to the expected offline optimum, called the \textit{Ratio of Expectations} (RoE). However, RoE has limitations as it only guarantees the average performance compared to the optimum, and might perform poorly against the realized ex-post optimal value. We study an alternative performance measure, the \textit{Expected Ratio} (EoR), namely the expectation of the ratio between algorithm's and prophet's value. EoR offers robust guarantees, e.g., a constant EoR implies achieving a constant fraction of the offline optimum with constant probability. For the special case of single-choice problems the EoR coincides with the well-studied notion of probability of selecting the maximum. However, the EoR naturally generalizes the probability of selecting the maximum for combinatorial constraints, which are the main focus of this paper. Specifically, we establish two reductions: for every constraint, RoE and the EoR are at most a constant factor apart. Additionally, we show that the EoR is a stronger benchmark than the RoE in that, for every instance (constraint and distribution), the RoE is at least a constant fraction of the EoR, but not vice versa. Both these reductions imply a wealth of EoR results in multiple settings where RoE results are known.