Time-Optimal Sublinear Algorithms for Matching and Vertex Cover

Abstract: We study the problem of estimating the size of maximum matching and minimum vertex cover in sublinear time. Denoting the number of vertices by $n$ and the average degree in the graph by $\bar{d}$, we obtain the following results for both problems: * A multiplicative $(2+\epsilon)$-approximation that takes $\tilde{O}(n/\epsilon<sup>2)$ time using adjacency list queries. * A multiplicative-additive $(2, \epsilon n)$-approximation in $\tilde{O}((\bar{d} + 1)/\epsilon<sup>2)$ time using adjacency list queries. * A multiplicative-additive $(2, \epsilon n)$-approximation in $\tilde{O}(n/\epsilon<sup>{3})$ time using adjacency matrix queries. All three results are provably time-optimal up to polylogarithmic factors culminating a long line of work on these problems. Our main contribution and the key ingredient leading to the bounds above is a new and near-tight analysis of the average query complexity of the randomized greedy maximal matching algorithm which improves upon a seminal result of Yoshida, Yamamoto, and Ito [STOC'09].