On Two-Pass Streaming Algorithms for Maximum Bipartite Matching

TL;DR

This paper investigates the limitations of two-pass streaming algorithms for maximum bipartite matching, establishing space complexity bounds and unifying existing algorithms through a meta-algorithm, highlighting the need for new techniques for further improvements.

Contribution

It proves space lower bounds for certain two-pass algorithms and introduces a meta-algorithm that unifies and optimizes existing approaches, showing their optimality within a class.

Findings

Space complexity lower bound of n^{1+Ω(1/log log n)} for (2/3+ε)-approximation algorithms.

Unification of two main techniques into a meta-algorithm that encompasses existing algorithms.

Identification of the optimality of Konrad's algorithm and discovery of a second optimal algorithm.

Abstract

We study two-pass streaming algorithms for Maximum Bipartite Matching (MBM). All known two-pass streaming algorithms for MBM operate in a similar fashion: They compute a maximal matching in the first pass and find 3-augmenting paths in the second in order to augment the matching found in the first pass. Our aim is to explore the limitations of this approach and to determine whether current techniques can be used to further improve the state-of-the-art algorithms. We give the following results: We show that every two-pass streaming algorithm that solely computes a maximal matching in the first pass and outputs a $(2/3+\epsilon)$-approximation requires $n^{1+\Omega(\frac{1}{\log \log n})}$ space, for every $\epsilon > 0$, where $n$ is the number of vertices of the input graph. This result is obtained by extending the Ruzsa-Szemer\'{e}di graph construction of [GKK, SODA'12] so as to ensure that the resulting graph has a close to perfect matching, the key property needed in our construction. This result may be of independent interest. Furthermore, we combine the two main techniques, i.e., subsampling followed by the Greedy matching algorithm [Konrad, MFCS'18] which gives a $2-\sqrt{2} \approx 0.5857$-approximation, and the computation of \emph{degree-bounded semi-matchings} [EHM, ICDMW'16][KT, APPROX'17] which gives a $\frac{1}{2} + \frac{1}{12} \approx 0.5833$-approximation, and obtain a meta-algorithm that yields Konrad's and Esfandiari et al.'s algorithms as special cases. This unifies two strands of research. By optimizing parameters, we discover that Konrad's algorithm is optimal for the implied class of algorithms and, perhaps surprisingly, that there is a second optimal algorithm. We show that the analysis of our meta-algorithm is best possible. Our results imply that further improvements, if possible, require new techniques.