We design new serial and parallel approximation algorithms for computing a maximum weight $b$-matching in an edge-weighted graph with a submodular objective function. This problem is NP-hard; the new algorithms have approximation ratio $1/3$, and are relaxations of the Greedy algorithm that rely only on local information in the graph, making them parallelizable. We have designed and implemented Local Lazy Greedy algorithms for both serial and parallel computers. We have applied the approximate submodular $b$-matching algorithm to assign tasks to processors in the computation of Fock matrices in quantum chemistry on parallel computers. The assignment seeks to reduce the run time by balancing the computational load on the processors and bounding the number of messages that each processor sends. We show that the new assignment of tasks to processors provides a four fold speedup over the currently used assignment in the NWChemEx software on $8000$ processors on the Summit supercomputer at Oak Ridge National Lab.
翻译:我们设计了新的序列算法和平行近似算法,用于在边缘加权图中计算最大重量($b$-匹配)和子模量目标函数。这是个问题;新算法是硬的;新算法是近似比率1/3美元,是贪婪算法的放松,仅依靠图中的地方信息,使其可以平行。我们设计并实施了用于系列计算机和平行计算机的本地Lazy贪婪算法。我们应用了近似亚模值($b$-匹配算法)来分配处理器在平行计算机量子化学中计算Fock矩阵时的任务。任务的目的是通过平衡处理器的计算负荷和将每个处理器发送的信息数捆绑来缩短运行时间。我们显示,对处理器的新任务指派为处理器提供了目前用于NWChemeEx软件的8000美元高级处理器在Oak Ridge National Lab的顶峰超级计算机上使用的四倍速度。