Online Matching with General Arrivals

• Main Authors: Gamlath, Buddhima, Kapralov, Michael, Maggiori, Andreas, Svensson, Ola, Wajc, David
• Format: Journal Article
• Language: English
• Published: 17-04-2019
• Subjects: Computer Science - Data Structures and Algorithms
• DOI: 10.48550/arxiv.1904.08255

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.