Complexity and Approximation for Discriminating and Identifying Code Problems in Geometric Setups

Abstract

We study geometric variations of the discriminating code problem. In the discrete version of the problem, a finite set of points P and a finite set of objects S are given in \(\mathbb {R}^d\). The objective is to choose a subset \(S^* \subseteq S\) of minimum cardinality such that for each point \(p_i \in P\), the subset \(S_i^* \subseteq S^*\) covering \(p_i\) satisfies \(S_i^*\ne \emptyset \), and each pair \(p_i,p_j \in P\), \(i \ne j\), we have \(S_i^* \ne S_j^*\). In the continuous version of the problem, 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 in P are placed on a horizontal line L, and the objects in S are finite-length line segments aligned with 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 one dimension. Still for the 1-dimensional discrete version, we design a polynomial-time 2-approximation algorithm. We also design a PTAS for both discrete and continuous versions in one dimension, for the restriction where 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-complete, and design polynomial-time approximation algorithms that produce \((16\cdot OPT+1)\)-approximate and \((64\cdot OPT+1)\)-approximate solutions respectively, using rounding of suitably defined integer linear programming problems. Finally, we apply our techniques to a related variant of the discrete problem, where instead of points and geometric objects we just have a set S of objects. The goal is to select a small subset \(S^*\) of objects so that all objects of S are discriminated by their intersection with the objects of \(S^*\). This problem can be viewed as a graph problem by stating it in terms of the vertices of the geometric intersection graph of S. Under this graph-theoretical form, it is known as the identifying code problem. We show that the identifying code problem for axis-parallel unit square intersection graphs (in \(d=2\)) can be solved in the same manner as for the discrete version of the discriminating code problem for unit square objects described above, and all our positive approximation results still hold in this setting.