Brief Announcement: Distributed Approximations for the Semi-matching Problem

Abstract

We consider the semi-matching problem in bipartite graphs. The network is represented by a bipartite graph G = (U ∪ V, E), where U corresponds to clients, V to servers, and E is the set of available connections between them. The goal is to find a set of edges M ⊆ E such that every vertex in U is incident to exactly one edge in M. The load of a server v ∈ V is defined as the square of its degree in M and the problem is to find an optimal semi-matching, i.e. a semi-matching that minimizes the sum of the loads of the servers. Formally, given a bipartite graph G = (U ∪ V,E), a semi-matching in G is a subgraph M such that deg _M (u) = 1 for every u ∈ U. A semi-matching M is called optimal if cost(M): = ∑  _v ∈ V (deg _M (v))² is minimal. It is not difficult to see that for any semi-matching M, $\tfrac{|U|^2}{|V|} \leq{\rm cost}(M) \leq \Delta |U|$ where Δ is such that max _v ∈ V d(v) ≤ Δ. Consequently, if M* is optimal and M is arbitrary, then ${\rm cost}(M) \leq \tfrac{\Delta |V| {\rm cost}(M*)}{|U|}.$ Our main result shows that in some networks the $\tfrac{\Delta |V|}{|U|}$ factor can be reduced to a constant (Theorem 1).

The research supported by grant N N206 565740.