eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
7:1
7:16
10.4230/LIPIcs.SoCG.2017.7
article
Faster Algorithms for the Geometric Transportation Problem
Agarwal, Pankaj K.
Fox, Kyle
Panigrahi, Debmalya
Varadarajan, Kasturi R.
Xiao, Allen
Let R, B be a set of n points in R^d, for constant d, where the points of R have integer supplies, points of B have integer demands, and the sum of supply is equal to the sum of demand. Let d(.,.) be a suitable distance function such as the L_p distance. The transportation problem asks to find a map tau : R x B --> N such that sum_{b in B}tau(r,b) = supply(r), sum_{r in R}tau(r,b) = demand(b), and sum_{r in R, b in B} tau(r,b) d(r,b) is minimized. We present three new results for the transportation problem when d(.,.) is any L_p metric:
* For any constant epsilon > 0, an O(n^{1+epsilon}) expected time randomized algorithm that returns a transportation map with expected cost O(log^2(1/epsilon)) times the optimal cost.
* For any epsilon > 0, a (1+epsilon)-approximation in O(n^{3/2}epsilon^{-d}polylog(U)polylog(n)) time, where U is the maximum supply or demand of any point.
* An exact strongly polynomial O(n^2 polylog n) time algorithm, for d = 2.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.7/LIPIcs.SoCG.2017.7.pdf
transportation map
earth mover's distance
shape matching
approximation algorithms