Dynamic Approximate Vertex Cover and Maximum Matching

Abstract

We consider the problem of maintaining a large matching or a small vertex cover in a dynamically changing graph. Each update to the graph is either an edge deletion or an edge insertion. We give the first randomized data structure that simultaneously achieves a constant approximation factor and handles a sequence of k updates in k. polylog(n) time. Previous data structures require a polynomial amount of computation per update. The starting point of our construction is a distributed algorithm of Parnas and Ron (Theor. Comput. Sci. 2007), which they designed for their sublinear-time approximation algorithm for the vertex cover size. This leads us to wonder whether there are other connections between sublinear algorithms and dynamic data structures.