The k-Colorable Unit Disk Cover Problem
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In this article, we consider colorable variations of the Unit Disk Cover (UDC) problem as follows. k-Colorable Discrete Unit Disk Cover (k-CDUDC): Given a set P of n points, and a set D of m unit disks (of radius=1), both lying in the plane, and a parameter k, the objective is to compute a set D'⊆ D such that every point in P is covered by at least one disk in D' and there exists a function χ:D'→ C that assigns colors to disks in D' such that for any d and d' in D' if d∩ d'≠∅, then χ(d)≠χ(d'), where C denotes a set containing k distinct colors. For the k-CDUDC problem, our proposed algorithms approximate the number of colors used in the coloring if there exists a k-colorable cover. We first propose a 4-approximation algorithm in O(m^7knlog k) time for this problem and then show that the running time can be improved by a multiplicative factor of m^k, where a positive integer k denotes the cardinality of a color-set. The previous best known result for the problem when k=3 is due to the recent work of Biedl et al., (2021), who proposed a 2-approximation algorithm in O(m^25n) time. For k=3, our algorithm runs in O(m^18n) time, faster than the previous best algorithm, but gives a 4-approximate result. We then generalize our approach to exhibit a O((1+⌈2/τ⌉)^2)-approximation algorithm in O(m^(⌊4π+8τ+τ^2/√(12)⌋)knlog k) time for a given 1 ≤τ≤ 2. We also extend our algorithm to solve the k-Colorable Line Segment Disk Cover (k-CLSDC) and k-Colorable Rectangular Region Cover (k-CRRC) problems, in which instead of the set P of n points, we are given a set S of n line segments, and a rectangular region R, respectively.
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