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, … , e_m , 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 2/3 (∼ .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 6/11 (∼ .545) , or 3/5 (=.6) in bipartite graphs. In this paper, we present an algorithm that computes a 2/3(∼ .66) -approximate matching using only O(nlog (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/loglog (n))) space; recent follow-up work by the same author improved this lower bound to 1+ln (2) ∼ .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.