Randomized Composable Coresets for Matching and Vertex Cover

Abstract

A common approach for designing scalable algorithms for massive data sets is to distribute the computation across, say k, machines and process the data using limited communication between them. A particularly appealing framework here is the simultaneous communication model whereby each machine constructs a small representative summary of its own data and one obtains an approximate/exact solution from the union of the representative summaries. If the representative summaries needed for a problem are small, then this results in a communication-efficient and \emph{round-optimal} (requiring essentially no interaction between the machines) protocol. Some well-known examples of techniques for creating summaries include sampling, linear sketching, and composable coresets. These techniques have been successfully used to design communication efficient solutions for many fundamental graph problems. However, two prominent problems are notably absent from the list of successes, namely, the maximum matching problem and the minimum vertex cover problem. Indeed, it was shown recently that for both these problems, even achieving a modest approximation factor of \polylog{(n)} requires using representative summaries of size \widetilde{\Omega}(n^2) i.e. essentially no better summary exists than each machine simply sending its entire input graph.