Approximating Semi-matchings in Streaming and in Two-Party Communication

Abstract

We study the streaming complexity and communication complexity of approximating unweighted semi-matchings. 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. [2003], 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 ⩽ ϵ ⩽ 1 uses space Õ( n 1+ϵ and computes an O( n (1−ϵ)/2 )-approximation to the semi-matching problem. Furthermore, with O(log n ) passes it is possible to compute an O(log n )-approximation with space Õ( n ). In the one-way two-party communication setting, we show that for every ϵ > 0, deterministic communication protocols for computing an O( n 1/(1+ϵ) c +1) -approximation require a message of size more than cn bits. We present two deterministic protocols communicating n and 2 n edges that compute an O√ n and an O(n 1/3 )-approximation, respectively. Finally, we improve on the results of Harvey et al. [2003] and prove new links between semi-matchings and matchings. While it was known that an optimal semi-matching contains a maximum matching, we show that there is a hierarchical decomposition of an optimal semi-matching into maximum matchings. A similar result holds for semi-matchings that do not admit length-two degree-minimizing paths.