Fast algorithms for (max, min)-matrix multiplication and bottleneck shortest paths

Given a directed graph with a capacity on each edge, the all-pairs bottleneck paths (APBP) problem is to determine, for all vertices s and t, the maximum flow that can be routed from s to t. For dense graphs this problem is equivalent to that of computing the (max, min)-transitive closure of a real-valued matrix. In this paper, we give a (max, min)-matrix multiplication algorithm running in time O(n(3+ω)/2) ≤ O(n2.688), where ω is the exponent of binary matrix multiplication. Our algorithm improves on a recent O(n2+ω/3) ≤ O(n2.792)- time algorithm of Vassilevska, Williams, and Yuster. Although our algorithm is slower than the best APBP algorithm on vertex capacitated graphs, running in O(n2.575) time, it is just as efficient as the best algorithm for computing the dominance product, a problem closely related to (max, min)-matrix multiplication. Our techniques can be extended to give subcubic algorithms for related bottleneck problems. The all-pairs bottleneck shortest paths problem (APBSP) asks for the maximum flow that can be routed along a shortest path. We give an APBSP algorithm for edge-capacitated graphs running in O(n(3+ω)/2) time and a slightly faster O(n2.657)-time algorithm for vertex-capactitated graphs. The second algorithm significantly improves on an O(n2.859)-time APBSP algorithm of Shapira, Yuster, and Zwick. Our APBSP algorithms make use of new hybrid products we call the distance-max-min product and dominance-distance product.

给定一个有向图，每条边都有一个容量，全对瓶颈路径（APBP）问题是对所有顶点s和t，确定从s到t可以路由的最大流。对于稠密图，这个问题等价于计算实值矩阵的（max，min）-传递闭包。本文给出了一个时间复杂度为O（n（3+ω）/2）≤ O（n2.688）的（max，min）-矩阵乘法算法，其中ω是二进制矩阵乘法的指数.我们的算法改进了Vassilevska，威廉姆斯和Yuster最近提出的O（n2+ω/3）≤ O（n2.792）时间的算法.虽然我们的算法比最好的APBP算法的顶点容量限制图，运行在O（n2.575）的时间，它是一样有效的最好的算法计算的优势产品，一个问题密切相关的（最大，最小）-矩阵乘法。 我们的技术可以扩展到相关的瓶颈问题的次立方算法。全对瓶颈最短路径问题（APBSP）要求沿最短路径沿着路由的最大流，我们给出了一个运行时间为O（n（3+ω）/2）的边容量图的APBSP算法和一个运行时间略快的O（n 2.657）的顶点容量图算法.第二个算法显着改善了O（n2.859）时间APBSP算法的Shapira，Yuster和Zwick。我们的APBSP算法利用新的混合产品，我们称之为距离最大最小产品和优势距离产品。