Online Stochastic Matching with Unknown Arrival Order: Beating $0.5$ against the Online Optimum
Enze Sun, Zhihao Gavin Tang, Yifan Wang
TL;DR
This work studies online stochastic bipartite matching with unknown arrival order, aiming to beat the $0.5$ barrier against the online optimum. The authors adapt order-competitive analysis, introducing a free–deterministic edge decomposition and a slackness LP to handle instances where the arrival order is not known, and they design a poly-time order-unaware algorithm achieving a constant improvement over $0.5$; the analysis splits into large-slackness and small-slackness regimes. The approach combines a baseline, non-adaptive mechanism with a free-deterministic pruning step, followed by a conditional LP-based branching that yields a $0.5+Ω(1)$ target; in the small-slack regime they obtain a $0.625- o(1)$-level bound via online rounding and a dynamic fractional solution. The results demonstrate that knowledge of the arrival order is not strictly necessary to surpass the $0.5$ threshold against the online optimum, highlighting meaningful beyond-worst-case gains in online optimization for matching problems. The methods provide new tools—slackness-driven branching, free-deterministic decomposition, and online submodular-welfare-inspired rounding—that may extend to broader online stochastic optimization problems beyond matching.
Abstract
We study the online stochastic matching problem. Against the offline benchmark, Feldman, Gravin, and Lucier (SODA 2015) designed an optimal $0.5$-competitive algorithm. A recent line of work, initiated by Papadimitriou, Pollner, Saberi, and Wajc (MOR 2024), focuses on designing approximation algorithms against the online optimum. The online benchmark allows positive results surpassing the $0.5$ ratio. In this work, adapting the order-competitive analysis by Ezra, Feldman, Gravin, and Tang (SODA 2023), we design a $0.5+Ω(1)$ order-competitive algorithm against the online benchmark with unknown arrival order. Our algorithm is significantly different from existing ones, as the known arrival order is crucial to the previous approximation algorithms.