Identification of points using disks

Abstract

We consider the problem of identifying n points in the plane using disks, i.e., minimizing the number of disks so that each point is contained in a disk and no two points are in exactly the same set of disks. This problem can be seen as an instance of the test covering problem with geometric constraints on the tests. We give tight lower and upper bounds on the number of disks needed to identify any set of n points of the plane. In particular, we prove that if there are no three colinear points nor four cocyclic points, then 2⌈n/6⌉ + 1 disks are enough, improving the known bound of ⌈(n + 1)/2⌉ when we only require that no three points are colinear. We also consider complexity issues when the radius of the disks is fixed, proving that this problem is NP-complete. In contrast, we give a linear-time algorithm computing the exact number of disks if the points are colinear.

Discriminating Codes in Geometric Setups

“…Here, the challenge is to overcome the linear nature of the problem and to transmit the information across the entire construction without affecting intermediate regions. This result is in contrast with Continuous-G-Min-Disc-Code in 1D, which is polynomial-time solvable Gledel and Parreau (2019). This is also in contrast with most geometric covering problems, which are usually polynomial-time solvable in 1D Krupa R. et al (2017).…”

“…In the context of the above-mentioned practical applications, Discrete-G-Min-Disc-Code in 2D was defined in Basu et al (2019), where an integer programming formulation (ILP) of the problem was given along with an experimental study. Continuous-G-Min-Disc-Code was introduced in Gledel and Parreau (2019), and shown to be NP-complete for disks in 2D, but polynomial-time in 1D (even when the intervals are restricted to have bounded length). These two problems are related to the class of geometric covering problems, for which also both the discrete and continuous version are studied extensively Krupa R. et al (2017).…”

“…We also study both problems in 2D for axis-parallel unit square objects, which form a natural extension of 1D intervals to the 2D setting. The continuous version is known to be NP-complete for unit disks Gledel and Parreau (2019), and we show that the reduction can be adapted to our setting, for both the continuous and discrete case. We then design polynomial-time constant-factor approximation algorithms for both problems in the same setting, of factors 4 + for Continuous-G-Min-Disc-Code, and 32 + for Discrete-G-Min-Disc-Code (for any fixed > 0).…”

“…It has been shown that Continuous-G-Min-Disc-Code is polynomial-time solvable in 1D Gledel and Parreau (2019). Thus, in this section we mainly focus on Discrete-G-Min-Disc-Code.…”

“…In Gledel and Parreau (2019), it was shown that Continuous-G-Min-Disc-Code for bounded-radius disks is NP-complete. The same proof technique, a reduction from the NP-complete P 3 -Partition-Grid problem van Bevern et al (2014), can be adapted to show the following.…”