Online Stochastic Max-Weight Bipartite Matching: Beyond Prophet Inequalities
Online Stochastic Max-Weight Bipartite Matching: Beyond Prophet Inequalities
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DOI：
10.1145/3465456.3467613
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发表时间：
2021-02
期刊：
影响因子：
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通讯作者：
C. Papadimitriou;Tristan Pollner;A. Saberi;David Wajc
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文献类型：
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作者：
C. Papadimitriou;Tristan Pollner;A. Saberi;David Wajc
关于在线贝叶斯选择问题的丰富文献长期以来一直关注所谓的先知不等式，它将在线算法的增益与知道未来的“先知”的增益进行比较。一个同样自然但研究较少的基准是最佳在线算法，它可能是万能的（即计算上无限制的），但不是万能的。在线最优的计算复杂度是多少？多项式时间算法能有多好地逼近它？受打车应用的启发，我们研究了顶点到达下的在线随机最大权重匹配问题的上述问题。这个问题最近由 Ezra、Feldman、Gravin 和 Tang (EC'20) 提出，他们为此给出了 1/2 竞争算法。这是最好的比率，因为这个问题是原始单项预言不等式的推广。我们提出了一种多项式时间算法，该算法在 0.51 倍内近似在线最优——击败了最佳可能的预言不等式。我们结果的核心是一个新的线性程序公式，一种尝试匹配两次尝试中到达的顶点的算法，以及限制第二次尝试产生的相关性的分析。相比之下，我们表明在某个常数 α < 1 内近似这个问题是 PSPACE 困难的。
The rich literature on online Bayesian selection problems has long focused on so-called prophet inequalities, which compare the gain of an online algorithm to that of a "prophet" who knows the future. An equally-natural, though significantly less well-studied benchmark is the optimum online algorithm, which may be omnipotent (i.e., computationally-unbounded), but not omniscient. What is the computational complexity of the optimum online? How well can a polynomial-time algorithm approximate it? Motivated by applications in ride hailing, we study the above questions for the online stochastic maximum-weight matching problem under vertex arrivals. This problem was recently introduced by Ezra, Feldman, Gravin and Tang (EC'20), who gave a 1/2-competitive algorithm for it. This is the best possible ratio, as this problem is a generalization of the original single-item prophet inequality. We present a polynomial-time algorithm which approximates optimal online within a factor of 0.51---beating the best-possible prophet inequality. At the core of our result are a new linear program formulation, an algorithm that tries to match the arriving vertices in two attempts, and an analysis that bounds the correlation resulting from the second attempts. In contrast, we show that it is PSPACE-hard to approximate this problem within some constant α < 1.