Online Vertex-Weighted Bipartite Matching: Beating 1-1/e with Random Arrivals

TL;DR

This paper presents a new weighted online bipartite matching algorithm that surpasses the 1-1/e competitive ratio barrier under random arrivals, using a novel two-dimensional gain sharing function within the primal-dual framework.

Contribution

It introduces a weighted ranking algorithm with a time-dependent gain sharing function, achieving a competitive ratio of 0.6534, the first to exceed 1-1/e in this setting.

Findings

Achieves a 0.6534 competitive ratio for weighted online bipartite matching.

First to surpass 1-1/e ratio under randomized primal-dual framework.

Demonstrates benefits of random arrivals in weighted matching scenarios.

Abstract

We introduce a weighted version of the ranking algorithm by Karp et al. (STOC 1990), and prove a competitive ratio of 0.6534 for the vertex-weighted online bipartite matching problem when online vertices arrive in random order. Our result shows that random arrivals help beating the 1-1/e barrier even in the vertex-weighted case. We build on the randomized primal-dual framework by Devanur et al. (SODA 2013) and design a two dimensional gain sharing function, which depends not only on the rank of the offline vertex, but also on the arrival time of the online vertex. To our knowledge, this is the first competitive ratio strictly larger than 1-1/e for an online bipartite matching problem achieved under the randomized primal-dual framework. Our algorithm has a natural interpretation that offline vertices offer a larger portion of their weights to the online vertices as time goes by, and each online vertex matches the neighbor with the highest offer at its arrival.