NDLI: Extremum Properties of Hexagonal Partitioning and the Uniform Distribution in Euclidean Location

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Haimovich, M. Magnanti, T. L.
Copyright Year	1988
Abstract	A zero-sum game with a maximizer that selects a point x in given polygon R in the plane and a minimizer that selects K points $c_1 ,c_2 , \cdots ,c_K $ in the plane is considered; the payoff is the Euclidean distance from x to the closest of the points $c_j $, or any monotonically nondecreasing function of this quantity. Lower and upper bounds on the value of the game are derived by considering, respectively, the maximizers strategy of selecting a uniformly distributed random point in R and the minimizers strategy of selecting K members of a (uniformly) randomly positioned grid of centers that induce a covering of R by K congruent regular hexagons. The analysis shows that these strategies are asymptotically optimal (for $K \to \infty $). For Euclidean location problems with uniformly distributed customers, the results imply that hexagonal partitioning of the region is asymptotically optimal and that the uniform distribution is asymptotically the worst possible.
Starting Page	50
Ending Page	64
Page Count	15
File Format	PDF
ISSN	08954801
DOI	10.1137/0401006
e-ISSN	10957146
Journal	SIAM Journal on Discrete Mathematics (SJDMEC)
Issue Number	1
Volume Number	1
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-08-08
Access Restriction	Subscribed
Subject Keyword	Inequalities and extremum problems Polyhedra and polytopes; regular figures, division of spaces planar partitioning Combinatorial optimization K-median Problem location games facility location
Content Type	Text
Resource Type	Article
Subject	Mathematics

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