Positive linear programs (LPs) model many graph and operations research problems. One can solve for a $(1+\epsilon)$-approximation for positive LPs, for any selected $\epsilon$, in polylogarithmic depth and near-linear work via variations of the multiplicative weight update (MWU) method. Despite extensive theoretical work on these algorithms through the decades, their empirical performance is not well understood. In this work, we implement and test an efficient parallel algorithm for solving positive LP relaxations, and apply it to graph problems such as densest subgraph, bipartite matching, vertex cover and dominating set. We accelerate the algorithm via a new step size search heuristic. Our implementation uses sparse linear algebra optimization techniques such as fusion of vector operations and use of sparse format. Furthermore, we devise an implicit representation for graph incidence constraints. We demonstrate the parallel scalability with the use of threading OpenMP and MPI on the Stampede2 supercomputer. We compare this implementation with exact libraries and specialized libraries for the above problems in order to evaluate MWU's practical standing for both accuracy and performance among other methods. Our results show this implementation is faster than general purpose LP solvers (IBM CPLEX, Gurobi) in all of our experiments, and in some instances, outperforms state-of-the-art specialized parallel graph algorithms.

翻译：正线性规划（LPs）可建模许多图论与运筹学问题。对于任意选定的$\epsilon$，通过乘法权重更新（MWU）方法的变体，可在多对数深度和近线性工作量内求解正线性规划的$(1+\epsilon)$近似解。尽管这些算法经过数十年广泛的理论研究，其实际性能仍尚未被充分理解。在本工作中，我们实现并测试了一种求解正线性规划松弛的高效并行算法，并将其应用于稠密子图、二分图匹配、顶点覆盖和支配集等图问题。我们通过一种新的步长搜索启发式方法加速算法。实现中采用稀疏线性代数优化技术，例如向量运算融合与稀疏格式使用。此外，我们设计了图关联约束的隐式表示。通过在Stampede2超级计算机上使用线程级OpenMP和MPI，我们展示了并行可扩展性。我们将此实现与上述问题的精确库及专用库进行比较，以评估MWU方法在准确性和性能方面相对于其他方法的实际地位。实验结果表明，在所有测试中，该实现均快于通用线性规划求解器（IBM CPLEX、Gurobi），并且在某些实例中优于最先进的专用并行图算法。