Title

Optimal facility location problem on polyhedral terrains using descending paths

Article Type

Research Article

Publication Title

Theoretical Computer Science

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.

First Page

68

Last Page

75

DOI

10.1016/j.tcs.2020.09.037

Publication Date

12-22-2020

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