4,102 research outputs found
A deterministic near-linear time approximation scheme for geometric transportation
Given a set of points for some
constant and a supply function such that , , and , the geometric transportation problem asks one to find a
transportation map such that
, , and the weighted sum of
Euclidean distances for the pairs is minimized. We present the first deterministic algorithm that
computes, in near-linear time, a transportation map whose cost is within a factor of optimal. More precisely, our algorithm runs in
time for any constant
. While a randomized time
algorithm was discovered in the last few years, all previously known
deterministic -approximation algorithms run in
time. A similar situation existed for geometric bipartite
matching, the special case of geometric transportation where all supplies are
unit, until a deterministic time -approximation algorithm was presented at STOC 2022. Surprisingly,
our result is not only a generalization of the bipartite matching one to
arbitrary instances of geometric transportation, but it also reduces the
running time for all previously known -approximation
algorithms, randomized or deterministic, even for geometric bipartite matching,
by removing the dependence on the dimension from the exponent in the
running time's polylog.Comment: 23 page
Preconditioning for the Geometric Transportation Problem
In the geometric transportation problem, we are given a collection of points P in d-dimensional Euclidean space, and each point is given a supply of mu(p) units of mass, where mu(p) could be a positive or a negative integer, and the total sum of the supplies is 0. The goal is to find a flow (called a transportation map) that transports mu(p) units from any point p with mu(p) > 0, and transports -mu(p) units into any point p with mu(p) < 0. Moreover, the flow should minimize the total distance traveled by the transported mass. The optimal value is known as the transportation cost, or the Earth Mover\u27s Distance (from the points with positive supply to those with negative supply). This problem has been widely studied in many fields of computer science: from theoretical work in computational geometry, to applications in computer vision, graphics, and machine learning.
In this work we study approximation algorithms for the geometric transportation problem. We give an algorithm which, for any fixed dimension d, finds a (1+epsilon)-approximate transportation map in time nearly-linear in n, and polynomial in epsilon^{-1} and in the logarithm of the total supply. This is the first approximation scheme for the problem whose running time depends on n as n * polylog(n). Our techniques combine the generalized preconditioning framework of Sherman, which is grounded in continuous optimization, with simple geometric arguments to first reduce the problem to a minimum cost flow problem on a sparse graph, and then to design a good preconditioner for this latter problem
A simple deterministic near-linear time approximation scheme for transshipment with arbitrary positive edge costs
We describe a simple deterministic near-linear time approximation scheme for
uncapacitated minimum cost flow in undirected graphs with real edge weights, a
problem also known as transshipment. Specifically, our algorithm takes as input
a (connected) undirected graph , vertex demands such that , positive edge costs , and a parameter . In time, it returns a flow such that the net flow out of each
vertex is equal to the vertex's demand and the cost of the flow is within a factor of optimal. Our algorithm is combinatorial and has no
running time dependency on the demands or edge costs.
With the exception of a recent result presented at STOC 2022 for polynomially
bounded edge weights, all almost- and near-linear time approximation schemes
for transshipment relied on randomization in two main ways: 1) to embed the
problem instance into low-dimensional space and 2) to randomly pick target
locations to send flow so nearby opposing demands can be satisfied. Our
algorithm instead deterministically approximates the cost of routing decisions
that would be made if the input were subject to a random tree embedding. To
avoid computing the vertex-vertex distances that an approximation
of this kind suggests, we also limit the available routing decisions using
distances explicitly stored in the well-known Thorup-Zwick distance oracle
Aeronautical engineering: A continuing bibliography with indexes, supplement 100
This bibliography lists 295 reports, articles, and other documents introduced into the NASA Scientific and Technical Information System in August 1978
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