Secretary Matching With Vertex Arrivals and No Rejections

TL;DR

This paper introduces new online algorithms for matching problems with no rejections, achieving improved competitive ratios in various models including bipartite, non-bipartite, and roommate matching scenarios.

Contribution

It provides the first algorithms with competitive ratios for online matching problems with no rejections across multiple models, improving previous bounds significantly.

Findings

Capacitated bipartite matching with capacity 2 achieved a 4.62-competitive ratio.

Non-bipartite online matching achieved a 3.34-competitive ratio.

Roommate matching with social welfare maximization achieved a 7.96-competitive ratio.

Abstract

Most prior work on online matching problems has been with the flexibility of keeping some vertices unmatched. We study three related online matching problems with the constraint of matching every vertex, i.e., with no rejections. We adopt a model in which vertices arrive in uniformly random order and the non-negative edge-weights are arbitrary. For the capacitated online bipartite matching problem, in which the vertices of one side of the graph are offline and those of the other side arrive online, we give a 4.62-competitive algorithm when the capacity of each offline vertex is 2. For the online general (non-bipartite) matching problem, where all vertices arrive online, we give a 3.34-competitive algorithm. We also study the online roommate matching problem (Huzhang et al. 2017), in which each room (offline vertex) holds 2 persons (online vertices). Persons derive non-negative additive utilities from their room as well as roommate. In this model, with the goal of maximizing the social welfare, we give a 7.96-competitive algorithm. This is an improvement over the 24.72 approximation factor in (Huzhang et al. 2017).