(Optimal) Online Bipartite Matching with Degree Information

TL;DR

This paper introduces a model for online bipartite matching with degree predictions, proposing a greedy algorithm that is optimal in certain stochastic models and performs well empirically on real-world graphs.

Contribution

It presents the MinPredictedDegree algorithm that leverages degree predictions, proving its optimality in specific stochastic models and demonstrating strong empirical performance.

Findings

MinPredictedDegree is optimal in the Chung-Lu-Vu stochastic model.

The competitive ratio of MinPredictedDegree is at least 0.7299 in the known i.i.d. model.

It performs nearly as well as maximum matching on power law graphs.

Abstract

We propose a model for online graph problems where algorithms are given access to an oracle that predicts (e.g., based on modeling assumptions or on past data) the degrees of nodes in the graph. Within this model, we study the classic problem of online bipartite matching, and a natural greedy matching algorithm called MinPredictedDegree, which uses predictions of the degrees of offline nodes. For the bipartite version of a stochastic graph model due to Chung, Lu, and Vu where the expected values of the offline degrees are known and used as predictions, we show that MinPredictedDegree stochastically dominates any other online algorithm, i.e., it is optimal for graphs drawn from this model. Since the "symmetric" version of the model, where all online nodes are identical, is a special case of the well-studied "known i.i.d. model", it follows that the competitive ratio of MinPredictedDegree on such inputs is at least 0.7299. For the special case of graphs with power law degree distributions, we show that MinPredictedDegree frequently produces matchings almost as large as the true maximum matching on such graphs. We complement these results with an extensive empirical evaluation showing that MinPredictedDegree compares favorably to state-of-the-art online algorithms for online matching.