文章基本信息

标题：Discriminating Codes in Geometric Setups
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作者：Sanjana Dey ; Florent Foucaud ; Subhas C. Nandy 等
期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
电子版ISSN：1868-8969
出版年度：2020
卷号：181
页码：1-16
DOI：10.4230/LIPIcs.ISAAC.2020.24
出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
摘要：We study two geometric variations of the discriminating code problem. In the discrete version, a finite set of points P and a finite set of objects S are given in â"^d. The objective is to choose a subset S^* âS S of minimum cardinality such that the subsets S_i^* âS S^* covering p_i, satisfy S_i^* â â^. for each i = 1,2,â¦, n, and S_i^* â S_j^* for each pair (i,j), i â j. In the continuous version, the solution set S^* can be chosen freely among a (potentially infinite) class of allowed geometric objects. In the 1-dimensional case (d = 1), the points are placed on some fixed-line L, and the objects in S are finite segments of L (called intervals). We show that the discrete version of this problem is NP-complete. This is somewhat surprising as the continuous version is known to be polynomial-time solvable. This is also in contrast with most geometric covering problems, which are usually polynomial-time solvable in 1D. We then design a polynomial-time 2-approximation algorithm for the 1-dimensional discrete case. We also design a PTAS for both discrete and continuous cases when the intervals are all required to have the same length. We then study the 2-dimensional case (d = 2) for axis-parallel unit square objects. We show that both continuous and discrete versions are NP-hard, and design polynomial-time approximation algorithms with factors 4 Îµ and 32 Îµ, respectively (for every fixed Îµ > 0).
关键词：Discriminating code; Approximation algorithm; Segment stabbing; Geometric Hitting set