Matroid Online Bipartite Matching and Vertex Cover

TL;DR

This paper introduces a matroid-based generalization of online bipartite matching and vertex cover problems, extending classical results and providing competitive algorithms that incorporate matroid constraints for more expressive and practical models.

Contribution

It proposes the Matroid Online Bipartite Matching model, extending classical results to this new setting with competitive algorithms and a novel primal-dual waterfilling technique.

Findings

Achieves $1-1/e$-competitive algorithms for the new model.

Extends classical algorithms to matroid constraints.

Provides a primal-dual waterfilling procedure for matroid settings.

Abstract

The Adwords and Online Bipartite Matching problems have enjoyed a renewed attention over the past decade due to their connection to Internet advertising. Our community has contributed, among other things, new models (notably stochastic) and extensions to the classical formulations to address the issues that arise from practical needs. In this paper, we propose a new generalization based on matroids and show that many of the previous results extend to this more general setting. Because of the rich structures and expressive power of matroids, our new setting is potentially of interest both in theory and in practice. In the classical version of the problem, the offline side of a bipartite graph is known initially while vertices from the online side arrive one at a time along with their incident edges. The objective is to maintain a decent approximate matching from which no edge can be removed. Our generalization, called Matroid Online Bipartite Matching, additionally requires that the set of matched offline vertices be independent in a given matroid. In particular, the case of partition matroids corresponds to the natural scenario where each advertiser manages multiple ads with a fixed total budget. Our algorithms attain the same performance as the classical version of the problems considered, which are often provably the best possible. We present $1-1/e$-competitive algorithms for Matroid Online Bipartite Matching under the small bid assumption, as well as a $1-1/e$-competitive algorithm for Matroid Online Bipartite Matching in the random arrival model. A key technical ingredient of our results is a carefully designed primal-dual waterfilling procedure that accommodates for matroid constraints. This is inspired by the extension of our recent charging scheme for Online Bipartite Vertex Cover.