Online Stochastic Matching: Beating 1-1/e

TL;DR

This paper introduces a novel online stochastic bipartite matching algorithm that surpasses the traditional 1-1/e approximation barrier, achieving a 0.67-approximation by leveraging dual offline solutions and the power of two choices.

Contribution

The paper presents the first online algorithm for stochastic bipartite matching that beats the 1-1/e approximation barrier, using a new approach based on disjoint offline solutions.

Findings

Achieves a 0.67-approximation ratio, surpassing the 1-1/e barrier.

Employs a novel method combining max flow solutions and the power of two choices.

Provides bounds on the optimal solution using a carefully constructed cut.

Abstract

We study the online stochastic bipartite matching problem, in a form motivated by display ad allocation on the Internet. In the online, but adversarial case, the celebrated result of Karp, Vazirani and Vazirani gives an approximation ratio of $1-1/e$. In the online, stochastic case when nodes are drawn repeatedly from a known distribution, the greedy algorithm matches this approximation ratio, but still, no algorithm is known that beats the $1 - 1/e$ bound. Our main result is a 0.67-approximation online algorithm for stochastic bipartite matching, breaking this $1 - {1/e}$ barrier. Furthermore, we show that no online algorithm can produce a $1-\epsilon$ approximation for an arbitrarily small $\epsilon$ for this problem. We employ a novel application of the idea of the power of two choices from load balancing: we compute two disjoint solutions to the expected instance, and use both of them in the online algorithm in a prescribed preference order. To identify these two disjoint solutions, we solve a max flow problem in a boosted flow graph, and then carefully decompose this maximum flow to two edge-disjoint (near-)matchings. These two offline solutions are used to characterize an upper bound for the optimum in any scenario. This is done by identifying a cut whose value we can bound under the arrival distribution.