# On Two-Pass Streaming Algorithms for Maximum Bipartite Matching

Christian Konrad, Kheeran K. Naidu
2021

This is my very first publication in the proceedings of the 24th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX 2021).

I am grateful to my parents for always supporting and believing in me. I am thankful to my friends and loved ones who have always been a great source of inspiration and challenge. And last but not least, I could not have done this without the excellent guidance and mentorship from my supervisor and co-author Christian.

#### 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-Szemerédi graph construction of [Goel et al., 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 degree-bounded semi-matchings [Esfandiari et al., ICDMW’16][Kale and Tirodkar, 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.

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