Discriminating codes in geometric setups

Sanjana Dey, Florent Foucaud, Subhas C. Nandy, Arunabha Sen

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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 Rd. The objective is to choose a subset S ⊆ S of minimum cardinality such that the subsets Si ⊆ S covering pi, satisfy Si 6= ∅ for each i = 1, 2, . . ., n, and Si =6 Sj for each pair (i, j), i =6 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 + ∊ and 32 + ∊, respectively (for every fixed ∊ > 0).

Original languageEnglish (US)
Title of host publication31st International Symposium on Algorithms and Computation, ISAAC 2020
EditorsYixin Cao, Siu-Wing Cheng, Minming Li
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Pages241-2416
Number of pages2176
ISBN (Electronic)9783959771733
DOIs
StatePublished - Dec 2020
Event31st International Symposium on Algorithms and Computation, ISAAC 2020 - Virtual, Hong Kong, China
Duration: Dec 14 2020Dec 18 2020

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume181
ISSN (Print)1868-8969

Conference

Conference31st International Symposium on Algorithms and Computation, ISAAC 2020
CountryChina
CityVirtual, Hong Kong
Period12/14/2012/18/20

Keywords

  • Approximation algorithm
  • Discriminating code
  • Geometric hitting set
  • Segment stabbing

ASJC Scopus subject areas

  • Software

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