We consider prophet inequalities under general downward-closed constraints. In a prophet inequality problem, a decision-maker sees a series of online elements and needs to decide immediately and irrevocably whether or not to select each element upon its arrival, subject to an underlying feasibility constraint. Traditionally, the decision-maker's expected performance has been compared to the expected performance of the prophet, i.e., the expected offline optimum. We refer to this measure as the Ratio of Expectations (or, in short, RoE). However, a major limitation of the RoE measure is that it only gives a guarantee against what the optimum would be on average, while, in theory, algorithms still might perform poorly compared to the realized ex-post optimal value. Hence, we study alternative performance measures. In particular, we suggest the Expected Ratio (or, in short, EoR), which is the expectation of the ratio between the value of the algorithm and the value of the prophet. This measure yields desirable guarantees, e.g., a constant EoR implies achieving a constant fraction of the ex-post offline optimum with constant probability. Moreover, in the single-choice setting, we show that the EoR is equivalent (in the worst case) to the probability of selecting the maximum, a well-studied measure in the literature. This is no longer the case for combinatorial constraints (beyond single-choice), which is the main focus of this paper. Our main goal is to understand the relation between RoE and EoR in combinatorial settings. Specifically, we establish a two-way black-box reduction: for every feasibility constraint, the RoE and the EoR are at most a constant factor apart. This implies a wealth of EoR results in multiple settings where RoE results are known.
翻译:在一个先知的不平等问题中,决策者看到了一系列在线元素,需要立即和不可撤销地决定是否在到达时选择每个元素,但必须受到基本的可行性限制。传统上,决策者的预期性能与先知的预期性能相比较,即预期离线最佳。我们称这一计量为期望值比值(或简而言之,RoE)。然而,RoE措施的一个主要限制是,它只能保证平均的最佳程度,而理论上,算法仍可能与实现的事后最佳值相比表现不佳。因此,我们研究替代性能措施。特别是,我们提出预期性比值(或简而言之,EoR),这是算法值和先知价值之间的预期性比值。这一计量产生理想的保证,例如,不断的EOFO标准意味着以常态确定前向离线的最佳程度,而我们总是有不断的概率。此外,在单选取的RE规则中,我们最接近于E的概率,我们选择了E的概率,在选择E-rocho标准中,最接近于E-cho的概率,我们选择了E-cour case case a core a colental strue is a pain a pain a strate deal decreal deal demin the made the made the made the made a made the made the made the made the made the made lexin the made a pro pro pro pro pro preal deqental decreal decreal decreal decreal decal dece a made.