Zwick's @math -approximation algorithm for the All Pairs Shortest Path (APSP) problem runs in time @math , where @math is the exponent of matrix multiplication and @math denotes the largest weight. This can be used to approximate several graph characteristics including the diameter, radius, median, minimum-weight triangle, and minimum-weight cycle in the same time bound. Since Zwick's algorithm uses the scaling technique, it has a factor @math in the running time. In this paper, we study whether APSP and related problems admit approximation schemes avoiding the scaling technique. That is, the number of arithmetic operations should be independent of @math ; this is called strongly polynomial. Our main results are as follows. - We design approximation schemes in strongly polynomial time @math for APSP on undirected graphs as well as for the graph characteristics diameter, radius, median, minimum-weight triangle, and minimum-weight cycle on directed or undirected graphs. - For APSP on directed graphs we design an approximation scheme in strongly polynomial time @math . This is significantly faster than the best exact algorithm. - We explain why our approximation scheme for APSP on directed graphs has a worse exponent than @math : Any improvement over our exponent @math would improve the best known algorithm for Min-Max Product In fact, we prove that approximating directed APSP and exactly computing the Min-Max Product are equivalent.
We present two new algorithms for solving the All Pairs Shortest Paths (APSP) problem for weighted directed graphs. Both algorithms use fast matrix multiplication algorithms.The first algorithm solves the APSP problem for weighted directed graphs in which the edge weights are integers of small absolute value in O(n2+μ) time, where μ satisfies the equation ω(1, μ, 1) = 1 + 2μ and ω(1, μ, 1) is the exponent of the multiplication of an n × nμ matrix by an nμ × n matrix. Currently, the best available bounds on ω(1, μ, 1), obtained by Coppersmith, imply that μ 0 is an error parameter and W is the largest edge weight in the graph, after the edge weights are scaled so that the smallest non-zero edge weight in the graph is 1. It returns estimates of all the distances in the graph with a stretch of at most 1 + e. Corresponding paths can also be found efficiently.