(Fractional) Online Stochastic Matching via Fine-Grained Offline Statistics

TL;DR

This paper advances online stochastic matching by developing fractional algorithms with improved competitive ratios for non-i.i.d. and i.i.d. arrivals, surpassing classical barriers and enhancing existing methods.

Contribution

It introduces new fractional algorithms for non-i.i.d. and i.i.d. online arrivals in stochastic matching, achieving better competitive ratios and first surpassing the 1-1/e barrier in non-i.i.d. settings.

Findings

Fractional algorithms with 0.718 and 0.731 competitive ratios for non-i.i.d. and i.i.d. arrivals.

Proven upper bound of 0.75 for fractional algorithms in non-i.i.d. case.

Integral algorithms with 0.666 and 0.704 competitive ratios for non-i.i.d. and i.i.d. arrivals.

Abstract

Motivated by display advertising on the internet, the online stochastic matching problem is proposed by Feldman, Mehta, Mirrokni, and Muthukrishnan (FOCS 2009). Consider a stochastic bipartite graph with offline vertices on one side and with i.i.d. online vertices on the other side. The algorithm knows the offline vertices and the distribution of the online vertices in advance. Upon the arrival of each online vertex, its type is realized and the algorithm immediately and irrevocably decides how to match it. In the vertex-weighted version of the problem, each offline vertex is associated with a weight and the goal is to maximize the total weight of the matching. In this paper, we generalize the model to allow non-identical online vertices and focus on the fractional version of the vertex-weighted stochastic matching. We design fractional algorithms that are $0.718$-competitive and $0.731$-competitive for non i.i.d. arrivals and i.i.d. arrivals respectively. We also prove that no fractional algorithm can achieve a competitive ratio better than $0.75$ for non i.i.d. arrivals. Furthermore, we round our fractional algorithms by applying the recently developed multiway online correlated selection by Gao et al. (FOCS 2021) and achieve $0.666$-competitive and $0.704$-competitive integral algorithms for non i.i.d. arrivals and i.i.d. arrivals. Our results for non i.i.d. arrivals are the first algorithms beating the $1-1/e \approx 0.632$ barrier of the classical adversarial setting. Our $0.704$-competitive integral algorithm for i.i.d. arrivals slightly improves the state-of-the-art $0.701$-competitive ratio by Huang and Shu (STOC 2021).