Discriminating code and set cover with k-bend paths

Abstract

In this paper, we explore geometric discriminating code and set cover problems with k-bend paths. We first demonstrate that both the discriminating code problem and the set cover problem with unit 0-bend paths are NP-hard. Additionally, we establish that the set cover problem is NP-hard with unit 1-bend paths restricted to type ⌜, where horizontal segments intersect a vertical line and vertical segments intersect a horizontal line. Furthermore, we show that the discriminating code problem remains NP-hard with unit 1-bend paths of type ⌜, where all vertical segments intersect a horizontal line. Finally, we provide approximation algorithms for these two problems, specifically for 0-bend paths of uniform length and for 1-bend paths of type ⌜, where all horizontal and vertical segments are of equal length.