Improved Bounds for Matching in Random-Order Streams

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.