A near-optimal sublinear-time algorithm for approximating the minimum vertex cover size

We give a nearly optimal sublinear-time algorithm for approximating the size of a minimum vertex cover in a graph G. The algorithm may query the degree deg(v) of any vertex v of its choice, and for each 1 ≤ i ≤ deg(v), it may ask for the i^th neighbor of v. Letting VC_opt(G) denote the minimum size of vertex cover in G, the algorithm outputs, with high constant success probability, an estimate VC(G) such that VC_opt(G) ≤ VC(G) ≤ 2VC_opt(G) + εn, where ε is a given additive approximation parameter. We refer to such an estimate as a (2, ε)-estimate. The query complexity and running time of the algorithm are Õ(d̄ ·poly(1/ε)), where d̄ denotes the average vertex degree in the graph. The best previously known sublinear algorithm, of Yoshida et al. (STOC 2009), has query complexity and running time O(d⁴/ε²) , where d is the maximum degree in the graph. Given the lower bound of Ω(d) (for constant ε) for obtaining such an estimate (with any constant multiplicative factor) due to Parnas and Ron (TCS 2007), our result is nearly optimal. In the case that the graph is dense, that is, the number of edges is Θ(n²), we consider another model, in which the algorithm may ask, for any pair of vertices u and v, whether there is an edge between u and v. We show how to adapt the algorithm that uses neighbor queries to this model and obtain an algorithm that outputs a (2, ε)-estimate of the size of a minimum vertex cover whose query complexity and running time are Õ(n)·poly(1/ε).