Online Stochastic Matching: Online Actions Based on Offline Statistics

TL;DR

This paper introduces an online algorithm for stochastic bipartite matching that leverages offline statistics to achieve a competitive ratio of 0.702, improving previous bounds and establishing new hardness results.

Contribution

The paper presents a novel online algorithm using Monte Carlo sampling to improve competitive ratios in stochastic matching and provides tighter hardness bounds.

Findings

Achieves a competitive ratio of 0.702 for the online stochastic matching problem.

Improves the known competitive ratio bounds from previous work.

Proves a new upper bound of 0.823 for online algorithms under the known distribution model.

Abstract

We consider the online stochastic matching problem proposed by Feldman et al. [FMMM09] as a model of display ad allocation. We are given a bipartite graph; one side of the graph corresponds to a fixed set of bins and the other side represents the set of possible ball types. At each time step, a ball is sampled independently from the given distribution and it needs to be matched upon its arrival to an empty bin. The goal is to maximize the number of allocations. We present an online algorithm for this problem with a competitive ratio of 0.702. Before our result, algorithms with a competitive ratio better than $1-1/e$ were known under the assumption that the expected number of arriving balls of each type is integral. A key idea of the algorithm is to collect statistics about the decisions of the optimum offline solution using Monte Carlo sampling and use those statistics to guide the decisions of the online algorithm. We also show that our algorithm achieves a competitive ratio of 0.705 when the rates are integral. On the hardness side, we prove that no online algorithm can have a competitive ratio better than 0.823 under the known distribution model (and henceforth under the permutation model). This improves upon the 5/6 hardness result proved by Goel and Mehta \cite{GM08} for the permutation model.