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 <=L are considered?". A more general problem is the APSP-AF (all-pair shortest path for all flows) problem, in which each edge has two weights - a length d and a capacity f, and a query asks about the shortest path from u to v where only edges of capacity >= 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.
Graph Theory
Approximation Algorithms
Combinatorial Optimization
Theory of computation~Design and analysis of algorithms
42:1-42:12
Regular Paper
Ran
Duan
Ran Duan
Institute for Interdisciplinary Information Sciences, Tsinghua University, China
R. Duan is supported by a China Youth 1000-Talent grant.
Hanlin
Ren
Hanlin Ren
Institute for Interdisciplinary Information Sciences, Tsinghua University, China
10.4230/LIPIcs.ICALP.2018.42
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Ran Duan and Hanlin Ren
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