Time-Optimal Sublinear Algorithms for Matching and Vertex Cover

TL;DR

This paper develops time-efficient sublinear algorithms for approximating maximum matching and minimum vertex cover in graphs, achieving near-optimal results with novel analysis of greedy algorithms.

Contribution

It introduces new sublinear algorithms with provably optimal time complexity for approximating matching and vertex cover, based on a refined analysis of greedy algorithms.

Findings

Achieves near-optimal $(2+)$-approximation in  O(n/^2) time.

Provides  O((ar{d}+1)/^2) time algorithms for approximations.

Improves analysis of greedy maximal matching, leading to optimal bounds.

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 $\tilde{O}(n/\epsilon^2)$ time using adjacency list queries. * A multiplicative-additive $(2, \epsilon n)$-approximation in $\tilde{O}((\bar{d} + 1)/\epsilon^2)$ time using adjacency list queries. * A multiplicative-additive $(2, \epsilon n)$-approximation in $\tilde{O}(n/\epsilon^{3})$ 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].