# On the Continuous Fermat-Weber Problem

@article{Fekete2005OnTC,
title={On the Continuous Fermat-Weber Problem},
author={S{\'a}ndor P. Fekete and Joseph B. M. Mitchell and Karin Beurer},
journal={ArXiv},
year={2005},
volume={cs.CG/0310027}
}
• Published 15 October 2003
• Mathematics, Computer Science
• ArXiv
We give the firstexact algorithmic study of facility location problems that deal with finding a median for acontinuum of demand points. In particular, we consider versions of the "continuousk-median (Fermat-Weber) problem" where the goal is to select one or more center points that minimize the average distance to a set of points in a demandregion. In such problems, the average is computed as an integral over the relevant region, versus the usual discrete sum of distances. The resulting facility…
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## References

SHOWING 1-10 OF 74 REFERENCES
On the continuous Weber and k-median problems (extended abstract)
• Computer Science, Mathematics
SCG '00
• 2000
This work gives the first exact algorithmic study of facility location problems that deal with finding a median for a continuum of demand points, and provides polynomial-time algorithms for various versions of the L1 1-median (Weber) problem.
Algebraic optimization: The Fermat-Weber location problem
• Mathematics, Computer Science
Math. Program.
• 1990
This work exhibits an explicit solution to the strong separation problem associated with the Fermat-Weber model and shows that anε-approximation solution can be constructed in polynomial time using the standard Ellipsoid Method.
A Multifacility Location Problem on Median Spaces
• V. Chepoi
• Computer Science, Mathematics
Discret. Appl. Math.
• 1996
A polynomial algorithm for locating II new facilities in the median space when there are k facilities already located is presented and the objective is to minimize the weighted sum of distances.
Locating Facilities on the Manhattan Metric with Arbitrarily Shaped Barriers and Convex Forbidden Regions
• Computer Science
Transp. Sci.
• 1989
This paper considers two planar facility location problems while employing the Manhattan travel metric, and establishes that the search for an optimal solution can be restricted to a finite set of easily identifiable points.
Rectilinear and polygonal p-piercing and p-center problems
• Computer Science, Mathematics
SCG '96
• 1996
We consider the p-pie~cing problem, in which we are given a collection of regions, and wish to determine whether there exists a set of p points that intersects each of the given regions. We give
Probabilistic analysis of geometric location problems
• Eitan Zemel
• Mathematics, Computer Science
Ann. Oper. Res.
• 1984
As a by-product, asymptotically optimal algorithms for the 2-dimensionalp-normk median problem and for the twin problems of minimizing the maximum number of vertices served by any center and similarly for maximizing the minimum are obtained.
On the Set of Optimal Points to the Weber Problem: Further Results
• Mathematics, Computer Science
Transp. Sci.
• 1994
The paper contains a different view of the problem: whereas Drezner and Goldman use algebraic-analytical approach, the authors use a geometrical approach which permits us to obtain more general results and also clarifies the geometric nature of theproblem.
Technical Note - Algorithms for Weber Facility Location in the Presence of Forbidden Regions and/or Barriers to Travel
• Mathematics, Computer Science
Transp. Sci.
• 1994
This work uses the concept of visibility to create a network with the location point as the source and use Dijkstra's algorithm to compute the shortest distance to all the other demand points and develops an algorithm for finding the optimal solution.
The conditional p‐center problem in the plane
• Mathematics
• 1993
An algorithm is given for the conditional p-center problem, namely, the optimal location of one or more additional facilities in a region with given demand points and one or more preexisting
Worst-Case and Probabilistic Analysis of a Geometric Location Problem
This work considers the problem of choosing K “medians” among n points on the Euclidean plane such that the sum of the distances from each of the n points to its closest median is minimized and presents two heuristics that produce arbitrarily good solutions with probability going to 1.