New Bounds on the Average Distance from the Fermat-Weber Center of a Planar Convex Body
ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
New bounds on the average distance from the Fermat-Weber center of a planar convex body
Discrete Optimization
Dividing a Territory Among Several Vehicles
INFORMS Journal on Computing
Dividing a Territory Among Several Facilities
INFORMS Journal on Computing
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We consider the problem of balancing the load among several service-providing facilities, while keeping the total cost low. Let D be the underlying demand region, and let p 1,…,p m be m points representing m facilities. We consider the following problem: Subdivide D into m equal-area regions R 1,…,R m , so that region R i is served by facility p i , and the average distance between a point q in D and the facility that serves q is minimal. We present constant-factor approximation algorithms for this problem, with the additional requirement that the resulting regions must be convex. As an intermediate result we show how to partition a convex polygon into m equal-area convex subregions so that the fatness of the resulting regions is within a constant factor of the fatness of the original polygon. In fact, we prove that our partition is, up to a constant factor, the best one can get if one’s goal is to maximize the fatness of the least fat subregion. We also discuss the structure of the optimal partition for the aforementioned load balancing problem: indeed, we argue that it is always induced by an additive-weighted Voronoi diagram for an appropriate choice of weights.