Improved algorithms for placing undesirable facilities

Abstract

We improve several existing algorithms for determining the location of one or more undesirable facilities amidst a set P of n demand points, under various constraints and distance functions. We assume that the demand points reside within some given bounded region R. Applying concepts and techniques from Computational Geometry, we provide efficient algorithms for the following problems: (1) Maxmin multi-facility location: Locate k undesirable facilities within R under the constraints that the smallest distance between each demand point and the facilities is at least a given r, and the distance between any two facilities is at least a given D . Under the L ∞ (L 1) norm we present efficient algorithms for any k, and under the L 2 norm we can locate efficiently two such facilities. In all cases, R is assumed to be an axis-parallel rectangle. (2) Minsum coverage: Given a set of weighted demand points contained in an axis-parallel rectangular region R, and given a smaller axis-parallel rectangle Q, place Q within R such that the sum of weights of the demand points contained in Q is minimized. Scope and Purpose Using tools from Computational Geometry we study two facility location problems that were previously studied by Brimberg and Mehrez (Location Sci. 2 (1) (1994) 11), and Drezner and Wesolowsky (Location Sci. 2 (2) (1994) 83). Geometric instances of facility location problems have attracted researchers from the Computational Geometry community, especially in the last few years. Computational Geometry (see, e.g., the textbook Computational Geometry—Algorithms and Applications, by de Berg et al., Berlin: Springer, 1997) deals with the efficient processing of spatial data and with geometric optimization, and thus techniques, algorithms, and data structures from this field can be effectively utilized for solving facility location problems of a geometric flavor.