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 and S * are finite sub-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