Distributed near-optimal matching

Abstract

In this paper, we consider the following distributed bipartite matching problems: Let G=(L, R; E) be a bipartite graph in which boys are part L (left nodes), and girls are part R (right nodes.) There is an edge (l _i, r _j) ∈ E iff boy l _i is interested in girl r _j. Every boy l _i will propose to a girl r _j among all those he is interested in, i.e., his adjacent right nodes in the bipartite graph G. If several boys propose to the same girl, only one of them will be accepted by the girl. We assume that none of the boys communicate with others. This matching problem is typical of distributed computing under incomplete information: Each boy only knows his own preference but none of the others. We first study the one-round matching problem: each boy proposes to at most one girl, so that the total number of girls receiving a proposal is maximized. If the maximum matching is M, then no deterministic algorithm can produce a matching of size not bounded by a constant, but a randomized algorithm achieves √M—and this is shown optimal by an adversary argument. If we allow many rounds in matching, with the senders learning from their failures, then the ratio is unbounded when the number of rounds is smaller than Δ, and becomes bounded (two) at the Δ-th round. In contrast, an extension of the one-round randomized algorithm establishes that there is no such discontinuity in the randomized case. This randomized result is also matched by an upper bound of asymptotically the same order.