Access to Data and Number of Iterations

In this article, we consider graph algorithms in models of computation where the space usage (random accessible storage, in addition to the read-only input) is sublinear in the number of edges m and the access to input is constrained. These questions arise in many natural settings, and in particular in the analysis of streaming algorithms, MapReduce or similar algorithms, or message passing distributed computing that model constrained parallelism with sublinear central processing. We focus on weighted nonbipartite maximum matching in this article. For any constant p > 1, we provide an iterative sampling-based algorithm for computing a (1 − ε)-approximation of the weighted nonbipartite maximum matching that uses O ( p /ε) rounds of sampling, and O ( n 1+1/ p ) space. The results extend to b -Matching with small changes. This article combines adaptive sketching literature and fast primal-dual algorithms based on relaxed Dantzig-Wolfe decision procedures. Each round of sampling is implemented through linear sketches and can be executed in a single round of streaming or two rounds of MapReduce. The article also proves that nonstandard linear relaxations of a problem, in particular penalty-based formulations, are helpful in reducing the adaptive dependence of the iterations.