Online Matching Meets Sampling Without Replacement

TL;DR

This paper provides the first theoretical analysis of sampling without replacement in online matching, demonstrating competitive guarantees and derandomization for both online bipartite and stochastic matching problems.

Contribution

It introduces a novel potential function analysis framework and proves competitive ratios, including breaking the $1-1/e$ barrier with a deterministic algorithm.

Findings

Sampling without replacement is at least 0.707-competitive in online stochastic matching.

A polynomial-time deterministic algorithm surpassing the $1-1/e$ barrier is developed.

Sampling without replacement is at least 0.513-competitive in online bipartite matching.

Abstract

Sampling without replacement is a natural online rounding strategy for converting fractional bipartite matching into an integral one. In Online Bipartite Matching, we can use the Balance algorithm to fractionally match each online vertex, and then sample an unmatched offline neighbor with probability proportional to the fractional matching. In Online Stochastic Matching, we can take the solution to a linear program relaxation as a reference, and then match each online vertex to an unmatched offline neighbor with probability proportional to the fractional matching of the online vertex's type. On the one hand, we find empirical evidence that online matching algorithms based on sampling without replacement outperform existing algorithms. On the other hand, the literature offers little theoretical understanding of the power of sampling without replacement in online matching problems. This paper fills the gap in the literature by giving the first non-trivial competitive analyses of sampling without replacement for online matching problems. In Online Stochastic Matching, we develop a potential function analysis framework to show that sampling without replacement is at least $0.707$-competitive. The new analysis framework further allows us to derandomize the algorithm to obtain the first polynomial-time deterministic algorithm that breaks the $1-\frac{1}{e}$ barrier. In Online Bipartite Matching, we show that sampling without replacement provides provable online correlated selection guarantees when the selection probabilities correspond to the fractional matching chosen by the Balance algorithm. As a result, we prove that sampling without replacement is at least $0.513$-competitive for Online Bipartite Matching.