Efficient Primal-Dual Graph Algorithms for MapReduce

Abstract

In this paper, we obtain improved algorithms for two graph-theoretic 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 $(1+\epsilon )$ approximation and takes $O({\log n\over \epsilon ^2})$ MapReduce iterations, each iteration having a shuffle size of $O(m)$ and a reducer size of $O(d_{max})$. Here $m$ is the number of edges, $n$ is the number of vertices, and $d_{max}$ is the maximum degree of a node. This dominates the previous best MapReduce algorithm, which provided a $(2+\delta )$-approximation in $O({\log n\over \delta })$ iterations, with each iteration having a total shuffle size of $O(m)$ and a reducer size of $O(d_{max})$.

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.

Ashish Goel: Supported in part by the DARPA xdata program, by grant #FA9550-12-1-0411 from the U.S. Air Force Office of Scientific Research (AFOSR) and the Defense Advanced Research Projects Agency (DARPA), and by NSF Award 0904325.

Kamesh Munagala: Supported by NSF grants CCF- 0745761, CCF-1348696, IIS-0964560, and IIS-1447554; and by grant W911NF-14-1- 0366 from the Army Research Office (ARO). Part of this work was done while the author was visiting Twitter, Inc.