发表机构
University of Macau（澳门大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对成本最小化先知不等式问题，研究有界支撑分布，给出分布已知时的近紧竞争比刻画，并设计分布未知算法达到相同保证，扩展至非IID设置得到渐近最优竞争比。
AI 中文摘要
我们研究成本最小化的先知不等式问题，在该问题中，决策者依次观察$n$个独立同分布（IID）的随机变量。每次观察后，决策者必须要么接受当前实现并停止，要么拒绝它并继续观察下一个变量。目标是最小化所选值。与经典的最大化设置不同，如果前$n-1$个值均未被接受，则必须选择最后一个实现。Esfandiari等人（SIDMA 2017）表明，该问题在一般情况下不存在常数竞争的在线算法，这促使后续工作研究受限分布类别，例如完全分布（Livanos和Mehta，SODA 2024）以及具有有界极值的分布（Livanos和Mehta，EC 2025）。在本工作中，我们关注具有有界支撑的分布。对于支撑在$[1,b]$上的分布，我们将在线算法的竞争比刻画为$b$和$n$两者的函数。我们同时考虑分布已知和分布未知的设置。在分布已知的情况下，我们提供了最优在线算法的近乎紧的刻画，表明其竞争比对于所有$n\geq 2$至多为$b^{(1-1/n)^n}$。更重要的是，我们设计了一种分布未知的算法，该算法达到相同的保证，使用仅依赖于$b$和$n$的预定阈值序列。此外，我们将分析扩展到非IID设置，其中我们表明一个简单的单阈值算法实现了渐近最优的竞争比$\Theta(\sqrt{b})$。
英文摘要
We study the cost-minimization prophet inequality problem, in which a decision-maker sequentially observes $n$ independent and identically distributed (IID) random variables. After each observation, the decision-maker must either accept the current realization and stop, or reject it and continue with the next variable. The goal is to minimize the selected value. Unlike the classical maximization setting, if none of the first $n-1$ values is accepted, the final realization must be selected. Esfandiari et al. (SIDMA 2017) showed that this problem does not admit constant-competitive online algorithms in general, which motivates subsequent work on restricted distribution classes, such as entire distributions (Livanos and Mehta, SODA 2024) and distributions with bounded extreme values (Livanos and Mehta, EC 2025). In this work, we focus on distributions with bounded support. For distributions supported on $[1,b]$, we characterize the competitive ratio of online algorithms as a function of both $b$ and $n$. We consider both distribution-aware and distribution-oblivious settings. In the distribution-aware case, we provide a nearly tight characterization of the optimal online algorithm, showing that its competitive ratio is at most $b^{(1-1/n)^n}$ for all $n\geq 2$. More significantly, we design a distribution-oblivious algorithm that achieves the same guarantee, with a sequence of predetermined thresholds that depend only on $b$ and $n$. Furthermore, we extend our analysis to the non-IID setting, where we show that a simple single-threshold algorithm attains an asymptotically optimal competitive ratio of $Θ(\sqrt{b})$.
