Online Matching Meets Sampling Without Replacement
Zhiyi Huang, Chui Shan Lee, Jianqiao Lu, Xinkai Shu
TL;DR
This paper proves that sampling without replacement is at least $0.513$-competitive for Online Bipartite Matching, and shows that sampling without replacement provides provable online correlated selection guarantees when the selection probabilities correspond to the fractional matching chosen by the Balance algorithm.
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.