On Two-Pass Streaming Algorithms for Maximum Bipartite Matching

We study two-pass streaming algorithms for \textsf{Maximum Bipartite Matching} (\textsf{MBM}). All known two-pass streaming algorithms for \textsf{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 \textsc{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

Maximum Matching via Maximal Matching Queries

We study approximation algorithms for Maximum Matching that are given access to the input graph solely via an edge-query maximal matching oracle. More specifically, in each round, an algorithm queries a set of potential edges and the oracle returns a maximal matching in the subgraph spanned by the query edges that are also contained in the input graph. This model is more general than the vertex-query model introduced by binti Khalil and Konrad [FSTTCS\u2720], where each query consists of a subset of vertices and the oracle returns a maximal matching in the subgraph of the input graph induced by the queried vertices. In this paper, we give tight bounds for deterministic edge-query algorithms for up to three rounds. In more detail: 1) As our main result, we give a deterministic 3-round edge-query algorithm with approximation factor 0.625 on bipartite graphs. This result establishes a separation between the edge-query and the vertex-query models since every deterministic 3-round vertex-query algorithm has an approximation factor of at most 0.6 [binti Khalil, Konrad, FSTTCS\u2720], even on bipartite graphs. Our algorithm can also be implemented in the semi-streaming model of computation in a straightforward manner and improves upon the state-of-the-art 3-pass 0.6111-approximation algorithm by Feldman and Szarf [APPROX\u2722] for bipartite graphs. 2) We show that the aforementioned algorithm is optimal in that every deterministic 3-round edge-query algorithm has an approximation factor of at most 0.625, even on bipartite graphs. 3) Last, we also give optimal bounds for one and two query rounds, where the best approximation factors achievable are 1/2 and 1/2 + ?(1/n), respectively, where n is the number of vertices in the input graph

Improved Weighted Matching in the Sliding Window Model

We consider the Maximum-weight Matching (MWM) problem in the streaming sliding window model of computation. In this model, the input consists of a sequence of weighted edges on a given vertex set $V$ of size $n$. The objective is to maintain an approximation of a maximum-weight matching in the graph spanned by the $L$ most recent edges, for some integer $L$, using as little space as possible. Prior to our work, the state-of-the-art results were a $(3.5+\varepsilon)$-approximation algorithm for MWM by Biabani et al. [ISAAC'21] and a $(3+\varepsilon)$-approximation for (unweighted) Maximum Matching (MM) by Crouch et al. [ESA'13]. Both algorithms use space $\tilde{O}(n)$. We give the following results: 1. We give a $(2+\varepsilon)$-approximation algorithm for MWM with space $\tilde{O}(\sqrt{nL})$. Under the reasonable assumption that the graphs spanned by the edges in each sliding window are simple, our algorithm uses space $\tilde{O}(n \sqrt{n})$. 2. In the $\tilde{O}(n)$ space regime, we give a $(3+\varepsilon)$-approximation algorithm for MWM, thereby closing the gap between the best-known approximation ratio for MWM and MM. Similar to Biabani et al.'s MWM algorithm, both our algorithms execute multiple instances of the $(2+\varepsilon)$-approximation $\tilde{O}(n)$-space streaming algorithm for MWM by Paz and Schwartzman [SODA'17] on different portions of the stream. Our improvements are obtained by selecting these substreams differently. Furthermore, our $(2+\varepsilon)$-approximation algorithm runs the Paz-Schwartzman algorithm in reverse direction over some parts of the stream, and in forward direction over other parts, which allows for an improved approximation guarantee at the cost of increased space requirements