Online Matching with KIID Edge Arrivals

2026-06-15

AI Summary

This study addresses online stochastic matching under the edge-arrival model, where edges of a general graph arrive independently according to a known distribution. The authors propose a two-stage algorithm that integrates greedy selection with suggested matchings. Under the KIID (known independent and identically distributed) assumption and leveraging the Natural LP analysis framework, this algorithm achieves the first competitive ratio exceeding the classical \(1 - 1/e\) barrier in the general-graph edge-arrival setting; notably, it strictly surpasses \(1 - 1/e\) when arrival rates are integral. The work also establishes that pure greedy algorithms are limited to a competitive ratio of at most 0.5, whereas suggested matchings alone can attain \(1 - 1/e\), thereby demonstrating the effectiveness of the proposed approach in harnessing prior distributional knowledge to enhance performance.

Abstract

In the classic online stochastic matching proposed by Feldman et al. (FOCS 2009), there is a known bipartite type-graph, where one side of the graph is given offline. Upon the arrival of each online vertex, its type is sampled independently and identically from the other side of the type-graph. This model has been extensively studied over the past decade, yielding a rich body of theoretical results. In this paper, we initiate the study of an edge arrival model for online stochastic matching. In our model, the online edges are sampled independently and identically (KIID) from a known type-graph, which need not be bipartite. We first show that the Greedy algorithm cannot achieve a competitive ratio strictly better than $0.5$ while the Suggested Matching algorithm has a competitive ratio of $1-1/e$ under the assumption of integral arrival rates, matching its performance in the one-sided vertex arrival model. We then propose a two-stage algorithm that combines Greedy and Suggested Matching, and show that its competitive ratio is strictly higher than $1-1/e$ for integral arrival rates. While our algorithm is simple, its analysis is intricate and builds upon the Natural LP, which has been proven very powerful in vertex arrival models. Our result reveals that even in the more challenging edge arrival setting for general graphs, competitive ratios better than $1-1/e$ are still possible, given the known distributions.

Problem

• online matching
• edge arrivals
• KIID
• competitive ratio
• stochastic matching

Innovation

• online stochastic matching
• edge arrivals
• competitive ratio
• two-stage algorithm
• Natural LP