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 $D=\{d_1,d_2,\cdots ,d_n\}$ of n demand points on the surface of a triangulated terrain $\mathcal{W}$ 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 $d\in D$ there exists a descending path from at least one facility $f\in F$ 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 $M\subseteq V$. The objective is to compute a set $C\subseteq V$ of minimum cardinality, such that for every node $v\in M$, either $v\in C$ or there exists a node $c\in C$ 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.

Introduction

De Berg and van Kreveld [1] started the study on the terrain navigation problem. They studied some height constrained path query problems on the surface of triangulated terrains. They presented a data structure to preprocess a triangulated terrain of m vertices in $O(m\log m)$ time and $O(m)$ space, so that for any two query points s and t, multiple path queries can be answered, each in $O(\log m)$ time, such as whether there is a descending path from s to t or whether there exists a path from s to t that stays below a given height. The main objective was efficient navigation on a terrain. Finally, they posed the question of finding a height constrained shortest path in $L_2$ (Euclidean norm) as an open problem.

More than a decade later, Ahmed et al. [2] presented two algorithms to construct a $(1+ϵ)$-approximate shortest descending path, SDP, for any given $ϵ\in (0,1)$. The running times are $O(\frac{n^{2}L}{ϵhcos\theta }\log (\frac{nL}{ϵhcos\theta }))$ and $O(\frac{n^{2}L}{ϵh}{\log }^2(\frac{nL}{ϵh}))$ respectively, where L is the largest edge length in the terrain, h is the smallest distance from a vertex to a non-incident edge in the same terrain face, and θ is the largest acute angle between a non-horizontal edge and a vertical line. Their algorithm is based on the idea of discretization of the terrain by adding Steiner points. Later, Cheng and Jin [3] presented an algorithm that computes a $(1+ϵ)$-approximate SDP in $O(n^4\log \frac{n}{ϵ})$ and thus, its running time does not depend on the geometric parameters of the terrain. They first compute the SDP in the $L_{\infty }$ metric using linear programming and then use it to get an approximation for the SDP in the $L_2$ metric. Later, Cheng and Jin [4] also presented an algorithm for computing the shortest paths on polyhedral surfaces and terrains under convex distance functions. In $O(n^2{\log }^{4}n)$ time, their algorithm can compute $L_1$ and $L_{\infty }$ shortest paths on a polyhedral surface. They also gave an $O(\frac{1}{\sqrt{ϵ}}{n}^2\log n+n^2{\log }^{4}n)$ time approximation algorithm that finds, for some $ϵ\in (0,1)$, $(1+ϵ)$-approximate shortest paths which have gradient constraints and cost functions imposed on them. The cost functions are linear combinations of path length and total ascent. Cheng et al. [5] presented a $(1+ϵ)$ factor approximation algorithm for the problem of finding the shortest paths in weighted regions in three dimensions.

Since the problem of finding SDP on the surface of general terrains seems to be hard, some interesting special cases have been considered. One such case is that of the convex terrains. Roy et al. [6] presented a $O(n^2\log n)$ time algorithm for convex terrains. Then, Roy [7] presented a faster algorithm that runs in $O(n{\log }^{2}n)$ time and this was improved by Wei and Joneja [8], who gave an $O(n\log n)$ time algorithm. An interesting open question here is to figure out whether this bound is tight for convex terrains. Ahmed and Lubiw [9] proposed a $(1+ϵ)$-factor approximation algorithm for finding an SDP through a given sequence of faces by formulating the problem as a convex optimization problem. Other special cases have also been considered in the papers cited above. For more information and a detailed survey on geodesic shortest paths, shortest paths in polygons and polyhedrons we refer the reader to chapter 31 of [10].

The objective of a facility location problem is the placement of a set of facilities that meet the demands of a distributed set of clients. A facility location problem typically deals with locating facilities as a subset of a given set of objects on a given environment to satisfy the demands of clients distributed on the same environment. Formally, given a weighted set D of demand points with weight distribution w, a set F of feasible facility locations with a non-negative cost distribution f and a distance function d that measures cost between a pair of locations, the objective is to find $F^{\prime }\subseteq F$, that optimizes some objective function $\mathcal{F}$ [11]. Constraints are placed on the nature of interactions between the demands and facilities and there may be constraints on the cost of placing the facilities and their maintenance. The constraints that must be satisfied can be of various types, including and not limited to minimizing the cost of placement and maintenance of the facilities, minimizing the overall distances from the clients to the facility/facilities, minimizing the time of delivery etc. This provides a company with many strategic advantages in taking decisions related to increasing profits. Facility location problems are widely studied in the fields of operation research and computer science. With the rise of the service economy, the interest in the field of facility location problems is only going to increase.

For a positive integer k, the k-centre and the k-median problems are known to be among the first studied problems in the field of facility location problems. Sylvester [12] posed the 1-centre problem in 1857. The 1-median problem was posed by Weber in his book [13] in the year 1909. The objective of the 1-centre problem is to decide the location of a facility on the plane such that the average time to reach a demand point, avoiding obstacles on the plane, is minimized. For $k\ge 1$, the problem is called the k-median problem. If the objective is to find a set $F^{\prime }$ of size k such that the maximum distance between a demand point and its nearest facility in the set $F^{\prime }$ is minimized, then the problem is called the k-centre problem. Aronov et al. [14] studied the problem of facility location on terrains. To compute the optimum placement they use the farthest-site Voronoi diagram. Their algorithm runs in $O(m{n}^2{\log }^{2}m(\log m+\log n))$ time, where n and m are the number of triangles and point-sites on the terrain. The field of facility location problems is vast and new variations arise frequently with practical considerations. We refer the reader to the books [15], [16], [17] for elaborate surveys on the field of facility location problems.

Motivated from the recent developments on descending path problem, we study a natural variation of the problem termed as descending facility location $(DFL)$ problem on the surface of a triangulated terrain $\mathcal{W}$, where the resources can be supplied from the facilities to any demand point through a descending path. A real world application of the DFL problem could be energy efficient resource distribution. Distribution of resources through descending paths could be energy efficient in real world and would require less fuel for transport. If the facilities are water reservoirs and the resource to be distributed is water, then no electricity would be required, as water flows through descending paths naturally.

DFL problem: Given a set D of n demand points, in a triangulated terrain $\mathcal{W}$ of m vertices, find a set of points $F\subset \mathcal{W}$ of minimum cardinality, such that, for any point in $d\in D$ there exists a descending path from at least one facility $f\in F$ to d. To the best of our knowledge, this is the first study of this kind.

Definition 1

A directed tree H is a graph such that, each edge of H has a unique direction and the underlying undirected graph of H is a tree.

The DFL problem can be reduced to the following graph theoretic problem, called the directed tree covering $(DTC)$ problem:

DTC problem: Given a directed tree $H=(V,E)$ and a set of marked nodes $M\subseteq V$, find a set $C\subseteq V$ of minimum cardinality, such that for every node $v\in M$, either $v\in C$ or there exists a node $c\in C$ such that v is reachable from c.

The general version of the DTC problem is the directed graph covering $(DGC)$ problem, where the input graph may be any directed graph, including one having cycles.

We show that the DFL problem can be solved in polynomial time. In Section 3, we prove that the DFL problem can be reduced to the DTC problem in $O((n+m)\log m)$ time, where n is the number of demand points and m is the number of vertices in the triangulated terrain. In the same section, we present an algorithm that solves the DTC problem in $O(|V|)$ time. Thereafter, we prove that the DFL problem can be solved in $O((n+m)\log m)$. In Section 4, we prove that the general version of the DTC problem, called the DGC problem, is NP-hard on directed bipartite graphs and hard to approximate within $(1-ϵ)\ln |M|$-factor, for all $ϵ>0$, 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.