Time-Optimal Sublinear Algorithms for Matching and Vertex Cover

We study the problem of estimating the size of maximum matching and minimum vertex cover in sub linear time. Denoting the number of vertices by$n$and the average degree in the graph by$\overline{d}$, we obtain the following results for both problems which are all provably time-optimal up to polylogarithmic factors:11The$\tilde{O}(\cdot)$notation hides polylog$n$factors throughout the paper. •A multiplicative$(2+\varepsilon)$-approximation that takes$\tilde{O}(n/\varepsilon^{2})$time using adjacency list queries. •A multiplicative-additive$(2,\ \varepsilon n)$-approximation that takes$\tilde{O}((\overline{d}+1)/\varepsilon^{2})$time using adjacency list queries. •A multiplicative-additive$(2,\ \varepsilon n)$-approximation that takes$\tilde{O}(n/\varepsilon^{3})$time using adjacency matrix queries. Our main contribution and the key ingredient of the bounds above is a near-tight analysis of the average query complexity of randomized greedy maximal matching which improves upon a seminal result of Yoshida, Yamamoto, and Ito$[\text{STOC}^{\prime} 09]$.