The Continuous 1.5D Terrain Guarding Problem: Discretization, Optimal Solutions, and PTAS

TL;DR

This paper addresses the NP-hard continuous 1.5D Terrain Guarding Problem by discretizing the terrain, proving NP-completeness, developing a PTAS, and creating an efficient IP-based algorithm for large terrains.

Contribution

It introduces a polynomial-size discretization for the continuous TGP, establishes NP-completeness, and provides a PTAS and an efficient algorithm for large instances.

Findings

Discretization reduces the problem to a manageable size.

NP-completeness of the continuous TGP is established.

An efficient IP-based algorithm solves large terrains within minutes.

Abstract

In the NP-hard continuous 1.5D Terrain Guarding Problem (TGP) we are given an $x$-monotone chain of line segments in $\mathbb{R}^2$ (the terrain $T$) and ask for the minimum number of guards (located anywhere on $T$) required to guard all of $T$. We construct guard candidate and witness sets $G, W \subset T$ of polynomial size such that any feasible (optimal) guard cover $G^* \subseteq G$ for $W$ is also feasible (optimal) for the continuous TGP. This discretization allows us to (1) settle NP-completeness for the continuous TGP, (2) provide a Polynomial Time Approximation Scheme (PTAS) for the continuous TGP using the PTAS for the discrete TGP by Gibson et al., and (3) formulate the continuous TGP as an Integer Linear Program (IP). Furthermore, we propose several filtering techniques reducing the size of our discretization, allowing us to devise an efficient IP-based algorithm that reliably provides optimal guard placements for terrains with up to $10^6$ vertices within minutes on a standard desktop computer.