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 e1, . . ., em, and the algorithm is allowed to make a single pass over this stream while using O(npolylog(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 ²₃ (∼ .66)-approximate matching, but the space requirement is O(n^1.5polylog(n)). Very recently, Farhadi et al. [SODA 2020] presented an algorithm with the desired space usage of O(npolylog(n)), but a worse approximation ratio of ₁₁⁶ (∼ .545), or ³₅ (= .6) in bipartite graphs. In this paper, we present an algorithm that computes a ²₃ (∼ .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 − ¹_e (∼ .63)-approximation requires (n^1+Ω(1/ log log(n⁾⁾) space. Our result for random-order streams is the first to go beyond the adversarial-order lower bound, thus establishing that computing a maximum matching is provably easier in random-order streams.