Matching with our Eyes Closed

TL;DR

This paper introduces a new randomized greedy algorithm for the query-commit matching problem, achieving a 56% approximation ratio and establishing bounds on the best possible performance of such algorithms.

Contribution

It proposes a novel randomized greedy algorithm with a 0.56 approximation factor and provides upper bounds for the performance of all randomized and vertex-iterative algorithms.

Findings

New randomized greedy algorithm attains at least 0.56 approximation.

No randomized algorithm can surpass 0.7916 approximation.

Vertex-iterative algorithms cannot exceed 0.75 approximation.

Abstract

Motivated by an application in kidney exchange, we study the following query-commit problem: we are given the set of vertices of a non-bipartite graph G. The set of edges in this graph are not known ahead of time. We can query any pair of vertices to determine if they are adjacent. If the queried edge exists, we are committed to match the two endpoints. Our objective is to maximize the size of the matching. This restriction in the amount of information available to the algorithm constraints us to implement myopic, greedy-like algorithms. A simple deterministic greedy algorithm achieves a factor 1/2 which is tight for deterministic algorithms. An important open question in this direction is to give a randomized greedy algorithm that has a significantly better approximation factor. This question was first asked almost 20 years ago by Dyer and Frieze [9] where they showed that a natural randomized strategy of picking edges uniformly at random doesn't help and has an approximation factor of 1/2 + o(1). They left it as an open question to devise a better randomized greedy algorithm. In subsequent work, Aronson, Dyer, Frieze, and Suen [2] gave a different randomized greedy algorithm and showed that it attains a factor 0.5 + epsilon where epsilon is 0.0000025. In this paper we propose and analyze a new randomized greedy algorithm for finding a large matching in a general graph and use it to solve the query commit problem mentioned above. We show that our algorithm attains a factor of at least 0.56, a significant improvement over 0.50000025. We also show that no randomized algorithm can have an approximation factor better than 0.7916 for the query commit problem. For another large and interesting class of randomized algorithms that we call vertex-iterative algorithms, we show that no vertex-iterative algorithm can have an approximation factor better than 0.75.