License: 
Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ISAAC.2020.24
URN: urn:nbn:de:0030-drops-133686
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/13368/
Dey, Sanjana ;
Foucaud, Florent ;
Nandy, Subhas C. ;
Sen, Arunabha
Discriminating Codes in Geometric Setups
Abstract
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 ℝ^d. The objective is to choose a subset S^* ⊆ S of minimum cardinality such that the subsets S_i^* ⊆ S^* covering p_i, satisfy S_i^* ≠ ∅ for each i = 1,2,…, n, and S_i^* ≠ S_j^* for each pair (i,j), i ≠ 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+ε and 32+ε, respectively (for every fixed ε > 0).
BibTeX - Entry
@InProceedings{dey_et_al:LIPIcs:2020:13368,
author = {Sanjana Dey and Florent Foucaud and Subhas C. Nandy and Arunabha Sen},
title = {{Discriminating Codes in Geometric Setups}},
booktitle = {31st International Symposium on Algorithms and Computation (ISAAC 2020)},
pages = {24:1--24:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-173-3},
ISSN = {1868-8969},
year = {2020},
volume = {181},
editor = {Yixin Cao and Siu-Wing Cheng and Minming Li},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2020/13368},
URN = {urn:nbn:de:0030-drops-133686},
doi = {10.4230/LIPIcs.ISAAC.2020.24},
annote = {Keywords: Discriminating code, Approximation algorithm, Segment stabbing, Geometric Hitting set}
}
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Keywords: |
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Discriminating code, Approximation algorithm, Segment stabbing, Geometric Hitting set |
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Collection: |
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31st International Symposium on Algorithms and Computation (ISAAC 2020) |
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Issue Date: |
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2020 |
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Date of publication: |
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04.12.2020 |
