Improved Bounds for Matching in Random-Order Streams

Abstract

We study the problem of computing an approximate maximum cardinality matching in the semi-streaming model when edges arrive in a random order. In the semi-streaming model, the edges of the input graph $G = (V,E)$ are given as a stream $e_1, \ldots , e_m$, and the algorithm is allowed to make a single pass over this stream while using $O(n\text {polylog}(n))$ space ($m = |E|$ and $n = |V|$). If the order of edges is adversarial, a simple single-pass greedy algorithm yields a 1/2-approximation in O(n) space; achieving a better approximation in adversarial streams remains an elusive open question. A line of recent work shows that one can improve upon the 1/2-approximation if the edges of the stream arrive in a random order. The state of the art for this model is two-fold: Assadi et al. [SODA 2019] show how to compute a $\frac{2}{3}$ $(\sim .66)$-approximate matching, but the space requirement is $O(n^{1.5}\text {polylog}(n))$. Very recently, Farhadi et al. [SODA 2020] presented an algorithm with the desired space usage of $O(n\text {polylog}(n))$, but a worse approximation ratio of $\frac{6}{11}$ $(\sim .545)$, or $\frac{3}{5}$ $(=.6)$ in bipartite graphs. In this paper, we present an algorithm that computes a $\frac{2}{3}(\sim .66)$-approximate matching using only $O(n\log (n))$ space, improving upon both results above. We also note that for adversarial streams, a lower bound of Kapralov [SODA 2013] shows that any algorithm that achieves a $1-\frac{1}{e}$($\sim .63$)-approximation requires $(n^{1+\Omega (1/\log \log (n))})$ space; recent follow-up work by the same author improved this lower bound to $1+\ln (2) \sim .59$ [SODA 2021]. As a consequence, both our result and the earlier result of Farhadi et al. prove that the problem of computing a maximum matching is strictly easier in random-order streams than in adversarial ones.