Space Lower Bounds for Approximating Maximum Matching in the Edge Arrival Model

The bipartite matching problem in the online and streaming settings has received a lot of attention recently. The classical vertex arrival setting, for which the celebrated Karp, Vazirani and Vazirani (KVV) algorithm achieves a $1-1/e$ approximation, is rather well understood: the $1-1/e$ approximation is optimal in both the online and semi-streaming setting, where the algorithm is constrained to use $n\cdot \log^{O(1)} n$ space. The more challenging the edge arrival model has seen significant progress recently in the online algorithms literature. For the strictly online model (no preemption) approximations better than trivial factor $1/2$ have been ruled out [Gamlath et al'FOCS'19]. For the less restrictive online preemptive model a better than $\frac1{1+\ln 2}$-approximation [Epstein et al'STACS'12] and even a better than $(2-\sqrt{2})$-approximation[Huang et al'SODA'19] have been ruled out. The recent hardness results for online preemptive matching in the edge arrival model are based on the idea of stringing together multiple copies of a KVV hard instance using edge arrivals. In this paper, we show how to implement such constructions using ideas developed in the literature on Ruzsa-Szemer\'edi graphs. As a result, we show that any single pass streaming algorithm that approximates the maximum matching in a bipartite graph with $n$ vertices to a factor better than $\frac1{1+\ln 2}\approx 0.59$ requires $n^{1+\Omega(1/\log\log n)}\gg n \log^{O(1)} n$ space. This gives the first separation between the classical one sided vertex arrival setting and the edge arrival setting in the semi-streaming model.