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A PTAS for the weighted unit disk cover problem. (English) Zbl 1440.68335
Halldórsson, Magnús M. (ed.) et al., Automata, languages, and programming. 42nd international colloquium, ICALP 2015, Kyoto, Japan, July 6–10, 2015. Proceedings. Part I. Berlin: Springer. Lect. Notes Comput. Sci. 9134, 898-909 (2015).
Summary: We are given a set of weighted unit disks and a set of points in Euclidean plane. The minimum weight unit disk cover (WUDC) problem asks for a subset of disks of minimum total weight that covers all given points. WUDC is one of the geometric set cover problems, which have been studied extensively for the past two decades (for many different geometric range spaces, such as (unit) disks, halfspaces, rectangles, triangles). It is known that the unweighted WUDC problem is NP-hard and admits a polynomial-time approximation scheme (PTAS). For the weighted WUDC problem, several constant approximations have been developed. However, whether the problem admits a PTAS has been an open question. In this paper, we answer this question affirmatively by presenting the first PTAS for WUDC. Our result implies the first PTAS for the minimum weight dominating set problem in unit disk graphs. Combining with existing ideas, our result can also be used to obtain the first PTAS for the maxmimum lifetime coverage problem and an improved constant approximation ratio for the connected dominating set problem in unit disk graphs.
For the entire collection see [Zbl 1316.68014].

##### MSC:
 68W25 Approximation algorithms 05C62 Graph representations (geometric and intersection representations, etc.) 05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.) 68U05 Computer graphics; computational geometry (digital and algorithmic aspects) 90C27 Combinatorial optimization
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