Efficient parallel implementations of approximation algorithms for guarding 1.5D terrains

Abstract

In the 1.5D terrain guarding problem, an x-monotone polygonal line is defined by k vertices and a G set of terrain points, i.e. guards, and a N set of terrain points which guards are to observe (guard). This involves a weighted version of the guarding problem where guards G have weights. The goal is to determine a minimum weight subset of G to cover all the points in N, including a version where points from N have demands. Furthermore, another goal is to determine the smallest subset of G, such that every point in N is observed by the required number of guards. Both problems are NP-hard and have a factor 5 approximation [3, 4]. This paper will show that if the $(1+\epsilon)$-approximate solver for the corresponding linear program is a computer, for any $\epsilon>0$, an extra $1+\epsilon$ factor will appear in the final approximation factor for both problems. A comparison will be carried out the parallel implementation based on GPU and CPU threads with the Gurobi solver, leading to the conclusion that the respective algorithm outperforms the Gurobi solver on large and dense inputs typically by one order of magnitude.