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

Abstract

We study the communication complexity and streaming 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. [WADS 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(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 ε > 0, deterministic communication protocols for computing an O$(n^{\frac{1}{(1+\epsilon)c + 1}})$-approximation require a message of size more than cn bits. We present two deterministic protocols communicating n and 2n edges that compute an O$(\sqrt{n})$ and an O(n ^1/3)-approximation respectively.

Finally, we improve on results of Harvey et al. [Journal of Algorithms 2006] 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 hierachical 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.

Research supported in part by ANR project RDAM. A full version of this paper can be found at: http://arxiv.org/abs/1304.6906