We present a method for solving the transshipment problem - also known as
uncapacitated minimum cost flow - up to a multiplicative error of 1+ε in undirected graphs with non-negative edge weights using a
tailored gradient descent algorithm. Using O~(⋅) to hide
polylogarithmic factors in n (the number of nodes in the graph), our gradient
descent algorithm takes O~(ε−2) iterations, and in each
iteration it solves an instance of the transshipment problem up to a
multiplicative error of polylogn. In particular, this allows
us to perform a single iteration by computing a solution on a sparse spanner of
logarithmic stretch. Using a randomized rounding scheme, we can further extend
the method to finding approximate solutions for the single-source shortest
paths (SSSP) problem. As a consequence, we improve upon prior work by obtaining
the following results: (1) Broadcast CONGEST model: (1+ε)-approximate SSSP using O~((n+D)ε−3) rounds, where D is the (hop) diameter of the network.
(2) Broadcast congested clique model: (1+ε)-approximate
transshipment and SSSP using O~(ε−2) rounds. (3)
Multipass streaming model: (1+ε)-approximate transshipment and
SSSP using O~(n) space and O~(ε−2) passes. The
previously fastest SSSP algorithms for these models leverage sparse hop sets.
We bypass the hop set construction; computing a spanner is sufficient with our
method. The above bounds assume non-negative edge weights that are polynomially
bounded in n; for general non-negative weights, running times scale with the
logarithm of the maximum ratio between non-zero weights.Comment: Accepted to SIAM Journal on Computing. Preliminary version in DISC
2017. Abstract shortened to fit arXiv's limitation to 1920 character