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 $\mathcal{NP}$-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.

我们研究稀疏矩阵二分法，即在并行稀疏矩阵-矢量乘法中最小化通信量的问题。通过图二等分的减少，我们证明了这个问题是$\mathcal{NP}$-如果分区的每一侧必须包含非零的线性分数，则为-complete。

我们提出一种改进的精确分支定界算法，该算法找到给定矩阵的最小通信量和最大允许的不平衡量。该算法基于最大流边界和打包边界，它们扩展了先前的匹配和打包边界。

我们在一个名为MP（矩阵分区程序）的新程序中实现了该算法，该程序可将SuiteSparse集合中的839个矩阵解析为最优，每个矩阵都在24小时的CPU时间内完成。此外，MP在大约3天的时间内解决了矩阵笼的难题。新程序平均比以前的程序MondriaanOpt快十倍以上。

使用839个最佳求解矩阵集进行的基准测试结果表明，将Mondriaan软件包的中粒度/迭代精化方法与PaToH软件包的超图二分法相结合，可以平均在最佳解决方案的10％之内生成稀疏矩阵二分。