Discriminating Codes in Geometric Setups
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).
Discriminating code
Approximation algorithm
Segment stabbing
Geometric Hitting set
Theory of computation~Approximation algorithms analysis
24:1-24:16
Regular Paper
A full version of the paper is available at https://arxiv.org/abs/2009.10353.
Sanjana
Dey
Sanjana Dey
ACM Unit, Indian Statistical Institute, Kolkata, India
Florent
Foucaud
Florent Foucaud
Univ. Bordeaux, Bordeaux INP, CNRS, LaBRI, UMR5800, 33400 Talence, France
This author was partially funded by the ANR project HOSIGRA (ANR-17-CE40-0022) and the IFCAM project "Applications of graph homomorphisms" (MA/IFCAM/18/39).
Subhas C.
Nandy
Subhas C. Nandy
ACM Unit, Indian Statistical Institute, Kolkata, India
Arunabha
Sen
Arunabha Sen
Arizona State University, Tempe, AZ, USA
10.4230/LIPIcs.ISAAC.2020.24
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Sanjana Dey, Florent Foucaud, Subhas C. Nandy, and Arunabha Sen
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