On the distributed complexity of the semi-matching problem

We consider the problem of matching clients with servers, each of which can process a subset of clients. It is known as the semi-matching or load balancing   problem in a bipartite graph G=(V,U,E)G=(V,U,E), where U corresponds to the clients and V   to the servers. The goal is to find a set of edges M⊆EM⊆E such that every vertex in U is incident to exactly one edge in M. The load   of a server v∈Vv∈V is defined as (degM⁡(v)+12), and the problem is to find a semi-matching M   that minimizes the sum of the loads of the servers. We show that to find an optimal solution in a distributed setting Ω(|V|)Ω(|V|) rounds are needed and propose distributed deterministic approximation algorithms for the problem. It yields 2-approximation and has time complexity O(Δ5)O(Δ5), where Δ is the maximum degree in V. We also give some greedy algorithms.