Discriminating codes in geometric setups

Document Type

Conference Article

Publication Title

Leibniz International Proceedings in Informatics, LIPIcs


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).

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