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

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 = vertical bar A vertical bar is a subset of edges S subset of 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 <= epsilon <= 1 uses space (O) over tilde (n(1+epsilon)) and computes an O(n((1-epsilon)/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 (O) over tilde (n). In the one-way two-party communication setting, we show that for every epsilon > 0, deterministic communication protocols for computing an O(n(1/(1+epsilon)c+ 1))-approximation require a message of size more than cn bits. We present two deterministic protocols communicating n and 2n edges that compute an O(root 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. Categories and Subject Descriptors: F.2.2. [Theory of Computation]: Analysis of Algorithms and Problem Complexity-Nonnumerical algorithms and problems