A Approximating Semi-matchings in Streaming and in Two-party Communication

We study the streaming complexity and communication complexity of approximating unweighted semimatchings. A semi-matching in a bipartite graph G = (A,B,E) with n = |A| is a subset of edges S ⊆ E that matches all A vertices to B vertices with the goal usually being to do this as fairly as possible. While the term semi-matching was coined in 2003 by Harvey et al. [WADS 2003, also Journal of Algorithms 2006], the problem had already previously been studied in the scheduling literature under different names. We present a deterministic one-pass streaming algorithm that for any 0 ≤ ≤ 1 uses space Õ(n1+ ) and computes an O(n(1− )/2)-approximation to the semi-matching problem. Furthermore, with O(logn) passes it is possible to compute an O(logn)-approximation with space Õ(n). In the one-way two-party communication setting, we show that for every > 0, deterministic commu-