发表机构
Georgia Tech, School of Industrial and Systems Engineering; University of Chicago, Booth School of Business; Stanford University, Department of Mathematics（佐治亚理工学院工业与系统工程学院; 芝加哥大学布斯商学院; 斯坦福大学数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对匹配体约束的先知不等式问题，提出两种在线到达模型下的新竞争比，并推广了现有拟阵结果，改进了已知的竞争比上界。
AI 中文摘要
在经典先知不等式问题中，算法以在线方式观察具有已知分布的随机变量序列，并且必须选择一个随机变量，以最大化其选择的期望值。算法的性能与一个“全知先知”进行比较，该先知在做出选择之前能观察到所有随机变量。经典先知不等式的组合扩展已被广泛研究，其中算法可以选择随机变量的一个子集（受限于属于某个可行集族）。我们在两种常见的在线到达模型下研究具有 $k$-匹配体约束（$k$-拟阵交约束和 $k$-有界超图匹配约束的常见推广）的先知不等式。我们针对全知先知的“事前”分数松弛给出保证，在对抗顺序情形下获得 $\frac{1}{k+1}$ 的事前竞争比，在随机顺序情形下获得 $\frac{1-e^{-k}}{k}$ 的竞争比。利用 Lee 和 Singla 的对偶框架（引用 \textit{Lee2018}），这也产生了这些设置下的在线冲突消解方案。我们的对抗顺序先知不等式可以看作是 Kalantarzadeh 和 Pashkovich 于 2026 年提出的最近 $\frac12$-竞争拟阵先知不等式的推广。该推广引入了一个新框架：跨多个拟阵的协调加权主划分。我们的随机顺序先知不等式是 Lee 和 Singla 于 2018 年提出的 $k=1$ 拟阵情形的推广。这两个结果分别改进了先前已知的 $k$-拟阵交的竞争比，先前分别为 $\frac{1}{(e+o(1))k}$ 和 $\frac{1}{k+1}$。
英文摘要
In the classical prophet inequality, an algorithm observes a sequence of random variables with known distributions in an online fashion, and it must select one of the random variables with the goal of maximizing the expected value of its selection. The performance of the algorithm is compared to an \textit{omniscient prophet} who observes all of the random variables before having to make its selection. Combinatorial extensions of the classical prophet inequality in which the algorithm gets to pick a subset of the random variables (constrained to belong to some family of feasible sets) have been studied extensively. We study prophet inequalities with a $k$-matchoid constraint (a common generalization of a $k$-matroid intersection constraint and a $k$-bounded hypergraph matching constraint) in two common online arrival models. We give guarantees with respect to the \textit{ex-ante} fractional relaxation of the omniscient prophet, obtaining an ex-ante competitive ratio of $\frac{1}{k+1}$ in the adversarial order case, and $\frac{1-e^{-k}}{k}$ in the random order case. Using the duality framework of Lee and Singla \cite{Lee2018}, this also yields online contention resolution schemes in these settings. Our adversarial-order prophet inequality can be viewed as a generalization of a recent $\frac12$-competitive matroid prophet inequality by Kalantarzadeh and Pashkovich, 2026. This generalization introduces a new framework: coordinated weighted principal partitions across multiple matroids. Our random-order prophet inequality is a generalization of the $k=1$ matroid case of Lee and Singla, 2018. The two results improve previously known competitive ratios for $k$-matroid intersection, which were $\frac{1}{(e+o(1))k}$ and $\frac{1}{k+1}$, respectively.
