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Discriminating Codes in Geometric Setups
- Source :
- 31st International Symposium on Algorithms and Computation (ISAAC 2020), 31st International Symposium on Algorithms and Computation (ISAAC 2020), Dec 2020, Hong-Kong, Hong Kong SAR China. ⟨10.4230/LIPIcs.ISAAC.2020.24⟩
- Publication Year :
- 2020
- Publisher :
- HAL CCSD, 2020.
-
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 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+ε and 32+ε, respectively (for every fixed ε > 0).<br />LIPIcs, Vol. 181, 31st International Symposium on Algorithms and Computation (ISAAC 2020), pages 24:1-24:16
Details
- Language :
- English
- Database :
- OpenAIRE
- Journal :
- 31st International Symposium on Algorithms and Computation (ISAAC 2020), 31st International Symposium on Algorithms and Computation (ISAAC 2020), Dec 2020, Hong-Kong, Hong Kong SAR China. ⟨10.4230/LIPIcs.ISAAC.2020.24⟩
- Accession number :
- edsair.doi.dedup.....20d0974b3ce63247fc59b23e1eba1e4e