Efficient Primal-Dual Algorithms for MapReduce

In this paper, we obtain improved algorithms for two graphtheoretic problems in the popular MapReduce framework. The first problem we consider is the densest subgraph problem. We present a primal-dual algorithm that provides a (1 + ) approximation and takes O( logn 2 ) MapReduce iterations, each iteration having a shuffle size of O(m) and a reduce-key-complexity of O(dmax). Here m is the number of edges, n is the number of vertices, and dmax is the maximum degree of a node. This dominates the previous best MapReduce algorithm, which provided a (2 + δ)-approximation in O( logn δ ) iterations, with each iteration having a total shuffle size of O(m) and a reduce-key-complexity of O(dmax). The standard primal-dual technique for solving the above problem results in O(n) iterations. Our key idea is to carefully control the width of the underlying polytope so that the number of iterations becomes small, but an approximate primal solution can still be recovered from the approximate dual solution. We then show an application of the same technique to the fractional maximum matching problem in bipartite graphs. Our results also map naturally to the PRAM model.