An improved exact algorithm and an NP-completeness proof for sparse matrix bipartitioning

Abstract

We investigate sparse matrix bipartitioning – a problem where we minimize the communication volume in parallel sparse matrix-vector multiplication. We prove, by reduction from graph bisection, that this problem is NP-complete in the case where each side of the bipartitioning must contain a linear fraction of the nonzeros.We present an improved exact branch-and-bound algorithm which finds the minimum communication volume for a given matrix and maximum allowed imbalance. The algorithm is based on a maximum-flow bound and a packing bound, which extend previous matching and packing bounds.We implemented the algorithm in a new program called MP (Matrix Partitioner), which solved 839 matrices from the SuiteSparse collection to optimality, each within 24 h of CPU-time. Furthermore, MP solved the difficult problem of the matrix cage6 in about 3 days. The new program is on average more than ten times faster than the previous program MondriaanOpt.Benchmark results using the set of 839 optimally solved matrices show that combining the medium-grain/iterative refinement methods of the Mondriaan package with the hypergraph bipartitioner of the PaToH package produces sparse matrix bipartitionings on average within 10% of the optimal solution.