The Query-commit Problem

TL;DR

This paper studies the query-commit problem, optimizing strategies for stochastic graph matching with applications in kidney exchanges, providing theoretical insights and demonstrating simple heuristics perform near-optimally in real-world scenarios.

Contribution

It introduces polynomial-time optimal strategies for sparse graphs, proves near-complete matching in large kidney exchange instances, and evaluates heuristics showing they perform close to optimal.

Findings

Optimal querying strategies for sparse graphs are polynomial-time computable.

Almost all nodes can be matched in large kidney exchange instances.

Simple heuristics perform within 1.5% of the optimal strategy.

Abstract

In the query-commit problem we are given a graph where edges have distinct probabilities of existing. It is possible to query the edges of the graph, and if the queried edge exists then its endpoints are irrevocably matched. The goal is to find a querying strategy which maximizes the expected size of the matching obtained. This stochastic matching setup is motivated by applications in kidney exchanges and online dating. In this paper we address the query-commit problem from both theoretical and experimental perspectives. First, we show that a simple class of edges can be queried without compromising the optimality of the strategy. This property is then used to obtain in polynomial time an optimal querying strategy when the input graph is sparse. Next we turn our attentions to the kidney exchange application, focusing on instances modeled over real data from existing exchange programs. We prove that, as the number of nodes grows, almost every instance admits a strategy which matches almost all nodes. This result supports the intuition that more exchanges are possible on a larger pool of patient/donors and gives theoretical justification for unifying the existing exchange programs. Finally, we evaluate experimentally different querying strategies over kidney exchange instances. We show that even very simple heuristics perform fairly well, being within 1.5% of an optimal clairvoyant strategy, that knows in advance the edges in the graph. In such a time-sensitive application, this result motivates the use of committing strategies.