# Online Matching with General Arrivals

## TL;DR

This paper investigates the power of randomization in online matching problems with general arrivals, showing it does not help for edge arrivals but improves performance for general vertex arrivals.

## Contribution

It resolves whether randomization can improve competitive ratios in general arrival models, establishing no benefit for edge arrivals and a positive benefit for vertex arrivals.

## Key findings

- Randomization does not improve the competitive ratio for edge arrivals.
- Randomized algorithms outperform deterministic ones for general vertex arrivals.
- The paper provides tight bounds for both models.

## Abstract

The online matching problem was introduced by Karp, Vazirani and Vazirani nearly three decades ago. In that seminal work, they studied this problem in bipartite graphs with vertices arriving only on one side, and presented optimal deterministic and randomized algorithms for this setting. In comparison, more general arrival models, such as edge arrivals and general vertex arrivals, have proven more challenging and positive results are known only for various relaxations of the problem. In particular, even the basic question of whether randomization allows one to beat the trivially-optimal deterministic competitive ratio of $\frac{1}{2}$ for either of these models was open. In this paper, we resolve this question for both these natural arrival models, and show the following. 1. For edge arrivals, randomization does not help --- no randomized algorithm is better than $\frac{1}{2}$ competitive. 2. For general vertex arrivals, randomization helps --- there exists a randomized $(\frac{1}{2}+\Omega(1))$-competitive online matching algorithm.