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

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