Sublinear Time Algorithm for Online Weighted Bipartite Matching

TL;DR

This paper introduces a sublinear time randomized algorithm for online weighted bipartite matching, enabling faster weight computation in large-scale systems without sacrificing matching quality.

Contribution

It provides a theoretical framework for approximate weight computation using randomized data structures, reducing complexity in practical online matching scenarios.

Findings

Weights can be computed in sublinear time with high probability.

The approximate weights preserve the competitive ratio of the matching algorithm.

The approach is applicable to large-scale recommendation and search systems.

Abstract

Online bipartite matching is a fundamental problem in online algorithms. The goal is to match two sets of vertices to maximize the sum of the edge weights, where for one set of vertices, each vertex and its corresponding edge weights appear in a sequence. Currently, in the practical recommendation system or search engine, the weights are decided by the inner product between the deep representation of a user and the deep representation of an item. The standard online matching needs to pay $nd$ time to linear scan all the $n$ items, computing weight (assuming each representation vector has length $d$), and then deciding the matching based on the weights. However, in reality, the $n$ could be very large, e.g. in online e-commerce platforms. Thus, improving the time of computing weights is a problem of practical significance. In this work, we provide the theoretical foundation for computing the weights approximately. We show that, with our proposed randomized data structures, the weights can be computed in sublinear time while still preserving the competitive ratio of the matching algorithm.