The Query-commit Problem

In the query-commit problem we are given a graph where edges have distinct prob-abilities of existing. It is possible to query the edges of the graph, and if the queriededge exists then its endpoints are irrevocably matched. The goal is to find a queryingstrategy which maximizes the expected size of the matching obtained. This stochasticmatching setup is motivated by applications in kidney exchanges and online dating.In this paper we address the query-commit problem from both theoretical andexperimental perspectives. First, we show that a simple class of edges can be queriedwithout compromising the optimality of the strategy. This property is then used toobtain 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 instancesmodeled over real data from existing exchange programs. We prove that, as the numberof 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 poolof patient/donors and gives theoretical justification for unifying the existing exchangeprograms. Finally, we evaluate experimentally different querying strategies over kidneyexchange instances. We show that even very simple heuristics perform fairly well, beingwithin 1.5% of an optimal clairvoyant strategy, that knows in advance the edges in thegraph. In such a time-sensitive application, this result motivates the use of committingstrategies.