Line-Constrained k-Semi-Obnoxious Facility Location
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Suppose we are given a set B of blue points and a set R of red points, all lying above a horizontal line ℓ, in the plane. Let the weight of a given point p_i∈ B∪ R be w_i>0 if p_i∈ B and w_i<0 if p_i∈ R, | B∪ R|=n, and d^0(=d∖∂ d) be the interior of any geometric object d. We wish to pack k non-overlapping congruent disks d_1, d_2, …, d_k of minimum radius, centered on ℓ such that ∑_j=1^k∑_{i:∃ p_i∈ R, p_i∈ d_j^0}w_i+∑_j=1^k∑_{i:∃ p_i∈ B, p_i∈ d_j}w_i is maximized, i.e., the sum of the weights of the points covered by ⋃_j=1^kd_j is maximized. Here, the disks are the obnoxious or undesirable facilities generating nuisance or damage (with quantity equal to w_i) to every demand point (e.g., population center) p_i∈ R lying in their interior. In contrast, they are the desirable facilities giving service (equal to w_i) to every demand point p_i∈ B covered by them. The line ℓ represents a straight highway or railway line. These k semi-obnoxious facilities need to be established on ℓ to receive the largest possible overall service for the nearby attractive demand points while causing minimum damage to the nearby repelling demand points. We show that the problem can be solved optimally in O(n^4k^2) time. Subsequently, we improve the running time to O(n^3k ·max(log n, k)). The above-weighted variation of locating k semi-obnoxious facilities may generalize the problem that Bereg et al. (2015) studied where k=1 i.e., the smallest radius maximum weight circle is to be centered on a line. Furthermore, we addressed two special cases of the problem where points do not have arbitrary weights.
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