Approximating All-Pair Bounded-Leg Shortest Path and APSP-AF in Truly-Subcubic Time

In the bounded-leg shortest path (BLSP) problem, we are given a weighted graph G with nonnegative edge lengths, and we want to answer queries of the form "what's the shortest path from u to v, where only edges of length = f are considered. In this article we give an O~(n^{(omega+3)/2}epsilon^{-3/2}log W) time algorithm to compute a data structure that answers APSP-AF queries in O(log(epsilon^{-1}log (nW))) time and achieves (1+epsilon)-approximation, where omega < 2.373 is the exponent of time complexity of matrix multiplication, W is the upper bound of integer edge lengths, and n is the number of vertices. This is the first truly-subcubic time algorithm for these problems on dense graphs. Our algorithm utilizes the O(n^{(omega+3)/2}) time max-min product algorithm [Duan and Pettie 2009]. Since the all-pair bottleneck path (APBP) problem, which is equivalent to max-min product, can be seen as all-pair reachability for all flow, our approach indeed shows that these problems are almost equivalent in the approximation sense.

[1]  Giuseppe F. Italiano,et al.  Dynamic shortest paths and transitive closure: Algorithmic techniques and data structures , 2006, J. Discrete Algorithms.

[2]  Tadao Takaoka,et al.  Combining All Pairs Shortest Paths and All Pairs Bottleneck Paths Problems , 2014, LATIN.

[3]  Raphael Yuster,et al.  All-pairs bottleneck paths for general graphs in truly sub-cubic time , 2007, STOC '07.

[4]  Ran Duan,et al.  Fast algorithms for (max, min)-matrix multiplication and bottleneck shortest paths , 2009, SODA.

[5]  Don Coppersmith,et al.  Matrix multiplication via arithmetic progressions , 1987, STOC.

[6]  Aaron Bernstein.  Maintaining Shortest Paths Under Deletions in Weighted Directed Graphs , 2016, SIAM J. Comput..

[7]  Giuseppe F. Italiano,et al.  Incremental algorithms for minimal length paths , 1991, SODA '90.

[8]  Ramesh Hariharan,et al.  Improved decremental algorithms for maintaining transitive closure and all-pairs shortest paths , 2007, J. Algorithms.

[9]  Michael Segal,et al.  On Bounded Leg Shortest Paths Problems , 2007, SODA '07.

[10]  Tadao Takaoka,et al.  Efficient Graph Algorithms for Network Analysis , 2013, ArXiv.

[11]  Monika Henzinger,et al.  Unifying and Strengthening Hardness for Dynamic Problems via the Online Matrix-Vector Multiplication Conjecture , 2015, STOC.

[12]  Ran Duan,et al.  Bounded-leg distance and reachability oracles , 2008, SODA '08.

[13]  Tadao Takaoka,et al.  Variations on the bottleneck paths problem , 2015, Theor. Comput. Sci..

[14]  Giri Narasimhan,et al.  Approximating Geometric Bottleneck Shortest Paths , 2003, STACS.

[15]  Uri Zwick,et al.  All pairs shortest paths in weighted directed graphs-exact and almost exact algorithms , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[16]  Daniel P. Martin.  Dynamic Shortest Path and Transitive Closure Algorithms: A Survey , 2017, ArXiv.

[17]  Tadao Takaoka,et al.  Combining the Shortest Paths and the Bottleneck Paths Problems , 2014, ACSC.