Online Matching with High Probability

TL;DR

This paper provides probabilistic bounds showing that the Ranking algorithm for online bipartite matching is highly likely to perform close to its expected competitive ratio, with extensions to related online matching problems.

Contribution

The paper establishes tail inequality bounds for the Ranking algorithm's competitive ratio, demonstrating high-probability performance guarantees in various online matching settings.

Findings

Ranking is (1 - 1/e - α)-competitive with high probability

Concentration results extend to Fully Online Matching and Vertex-Weighted Bipartite Matching

Probabilistic bounds improve understanding of algorithm performance variability

Abstract

We study the classical, randomized Ranking algorithm which is known to be $(1 - \frac{1}{e})$-competitive in expectation for the Online Bipartite Matching Problem. We give a tail inequality bound, namely that Ranking is $(1 - \frac{1}{e} - \alpha)$-competitive with probability at least $1 - e^{-2 \alpha^2 n}$ where $n$ is the size of the maximum matching in the instance. Building on this, we show similar concentration results for the Fully Online Matching Problem and for the Online Vertex-Weighted Bipartite Matching Problem.