Optimal facility location problem on polyhedral terrains using descending paths

Abstract

We study the descending facility location (DFL) problem on the surface of a triangulated terrain. A path from a point s to a point t on the surface of a terrain is descending if the heights of the subsequent points along the path from s to t are in a monotonically non-increasing order [1]. We are given a set D={d1,d2,⋯,dn} of n demand points on the surface of a triangulated terrain W and our objective is to find a set F (of points), of minimum cardinality, on the surface of the terrain such that for each demand point d∈D there exists a descending path from at least one facility f∈F to d. We present an O((n+m)log⁡m) time algorithm for solving the DFL problem, where m is the number of vertices in the triangulated terrain. We achieve this by reducing the DFL problem to a graph problem called the directed tree covering(DTC)problem. In the DTC problem, we have a directed tree B=(V,E) with a set of marked nodes M⊆V. The objective is to compute a set C⊆V of minimum cardinality, such that for every node v∈M, either v∈C or there exists a node c∈C such that v is reachable from c. We prove that the DFL problem can be reduced to DTC problem in O((m+n)log⁡m) time. The DTC problem thereafter can be solved in O(|V|) time. We also prove that the general version of the DTC problem, called the directed graph covering(DGC)problem is NP-hard on directed bipartite graphs and hard to approximate within (1−ϵ)ln⁡|M|-factor, for every ϵ>0, where |M| is the size of the set of marked nodes. We also prove that for the DGC problem, an O(log⁡|M|) factor approximation is possible and this approximation factor is tight.