A new bound and an O(mn) algorithm for the undesirable 1-median problem (maxian) on networks

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Abstract

The problem of locating an undesirable facility on a network with n nodes and m edges so as to maximize its total weighted distance to all nodes is addressed. We propose a new upper bound to the problem. Likewise, we develop a new algorithm in O(mn) time which dynamically updates this new upper bound. Computational results on low and high dense networks, as well as planar networks, are presented.

Introduction

Modern network location theory was originally introduced by Hakimi [1] in 1964. It basically deals with finding an optimal point on the network where one or more facilities can be established, so that the service demand of users (nodes) is completely satisfied. Usually, the facilities to be located are considered “desirable” for the customers, for instance, shopping centers, emergency services, schools, etc.

However, there are some services which are not so desirable, and might be considered annoying (obnoxious), such as garbage dump sites, oil plants or prisons. Some of them might be even harmful (noxious) for the surrounding population, for instance, nuclear reactors, chemical industries and polluting plants. Anyhow, we just consider all of them undesirable.

The literature on undesirable network location began in the mid 1970s with Church and Garfinkel [2], who defined and solved the 1-maxisum (maxian) problem in O(mnlogn) time, being n the number of nodes and m the number of edges. Later, Minieka [3] addressed the anti-center (maxmax) and the anti-median (maxsum), which is a similar approach to the unweighted case described in [2]. Soon after, Ting [4] developed an algorithm in linear time for the 1-maxisum problem on trees.

Regarding the maximin problem, Tamir [5] hinted at a method in O(mn) time using Megiddo [6] and Dyer [7]. Lately, Melachrinoudis and Zhang [8] and Berman and Drezner [9] make use of, respectively, [7] and [6], to devise analogous algorithms. The most recent approach is addressed by Colebrook et al. [10]. For a deeper and state-of-the-art survey on undesirable location, the reader is referred to Erkut and Neuman [11].

Tamir [12] briefly suggested that the 1-maxisum problem could be solved in O(mn) time using an algorithm given by Zemel [13]. However, to the best of our knowledge, there is no reference in the literature directly describing such an algorithm for the network 1-maxisum problem thus far. In this paper we present a new algorithm which solves this problem in O(mn) time.

We assume that the undesirableness uses the shortest paths from the facility to the customers, as Church and Garfinkel [2] considered in their paper. Accordingly, if traveling through a certain shortest path could turn dangerous due to traffic congestion, high density of population, unpleasant pollution propagation, etc, then these forbidden edges could be removed from the original network. Thus, provided that the new network is still connected, we can look for an alternative path.

The remainder of the paper is structured as follows. First, we introduce the notation and the general properties of the 1-maxisum problem. Section 3 addresses a new approach to the problem based on the right and left slopes of the objective function at the end nodes of each edge. In Section 4 we propose a new upper bound to this problem which speeds up the search of the optimal point. The next two sections describe, respectively, the new method and the O(mn) algorithm. In Section 7 a small trace is developed. Finally, we present the computational experience on low and high dense networks as well as planar networks. The paper ends with the conclusions and the discussion.

Section snippets

Notation and general properties

Let N=(V,E) be a simple (no loops or multiple edges), undirected, finite and connected network with n nodes (vertices) V={v1,v2,…,vn}, and m edges E={(vs,vt):vs,vt∈V}, with |E|=m. A function w:V→R, w(vi)=wi⩾0 is defined, which denotes the number of customers situated at vi who will make use of the facility's services. Obviously, we assume that not all wi=0. On the other hand, we define a function l:E→R+, l(e)=le>0 that indicates the length of edge e. Thus, a point x∈e ranges in the interval [0,l

A new approach

Given an edge e=(vs,vt)∈E, and for all nodes vi∈V, let di=d(vt,vi)−d(vs,vi) be the difference of weighted distances from nodes vs and vt to node vi. Obviously, from (2) we have −le⩽di⩽le. Using di we have bi=(di+le)/2. In particular, for di=−le, we get bi=0=vs, whereas for di=le, we obtain bi=le=vt. We define the following sets:A={vi∈V:−le<di⩽le},B={vi∈V:di=−le},C={vi∈V:−le⩽di<le},D={vi∈V:di=le}.Note that B⊆C, D⊆A and A∪B=C∪D=V.

Let W=∑vi∈Vwi be the sum of all the weights, and let Ws be the

Lower and upper bounds

On any edge e=(vs,vt)∈E, a simple lower bound LB(e)=max(F(vs),F(vt)) was proposed by Church and Garfinkel [2]. They also gave an upper bound for the unweighted maxian problem, which can be used to derive an initial upper bound for the weighted maxian problem as follows:UB(e)=F(vs)+F(vt)+Wle2.This bound is computed in O(n) time. However, this upper bound can be improved with the same time complexity as follows.

We consider both Ws and Wt to be strictly positive. Now, we compute the intersection

The method proposed when Ws and Wt are strictly positive

Church and Garfinkel [2] devised an O(mnlogn) algorithm to solve the maxian problem. Theorem 1 provides directly the solution when Ws or Wt is nonpositive. In this section we show how to obtain the optimal points in O(mn) time when Ws and Wt are strictly positive. Let e=(vs,vt)∈E. We begin by replacing (1) in (3) to get F(x)=∑vi∈Vwimin{x+d(vs,vi),le−x+d(vt,vi)}.

Given a point x on e, the following two sets are defined: L(x)={vi∈V:bi<x} and R(x)={vi∈V:bi⩾x}. The set L(x) contains the nodes with

The new algorithm

In this section, we bring together all the results previously stated. First, we outline the new algorithm and then we prove its complexity.

function NewAlgorithm(Network N, DistanceMatrix d)
{//FN: Current best value on network N,S: Solution set.
FN≔0,S≔∅
for all edges e=(vs,vt)∈E do
{  Compute Ws and Wt by ,
//Let Xe represent either a single point x or an interval [x1,x2].
Xe≔∅
if Theorem 1 holds then Store solution in Xe
else
{Fj≔F(vs),Wj≔Ws
Fk≔F(vt),Wk≔Wt
//Compute initial value of the new upper bound.

An example

Consider the network in Fig. 5, with n=7 nodes and m=15 edges. The node weights (in bold) are integers randomly generated between 1 and 9, whereas the edge lengths range between 1 and 25. The total weight W is equal to 24. Table 1 summarizes the trace.

We begin with edge (v1,v3). The slopes of function F(x) at nodes vs and vt are, respectively, Ws=20 and Wt=18. Thus, the new upper bound is NUB(e)=521.526. We have also included in the table the upper bound UB(e) and the difference between this

Computational results

The computational experience was performed using C++ programming language (GNU g++ 2.95.2) and LEDA (Library of Efficient Datatypes and Algorithms [15]), on a DEC with four alpha 466MHz processors and 2Gb of RAM, running OSF Digital UNIX.

Despite Tamir [12] having briefly stated that a solution to the network 1-maxisum problem can be obtained in O(mn) time using the general algorithms proposed by Zemel [13], the procedure is not directly described. As a result of this, we decided to compare the

Conclusions and discussion

The 1-maxisum location problem (maxian) problem on networks is analyzed. From [2], an initial upper bound UB(e) is derived, which is improved with a new upper bound NUB(e). Likewise, this bound can be dynamically updated without increasing the total computational time.

We have developed a new algorithm in O(mn) to solve this problem. The procedure makes use of the new upper bound, and hence, allows skipping out from the search process as soon as the upper bound is less than the global optimum.

Acknowledgements

The authors wish to thank the two anonymous reviewers for their valuable comments and suggestions. This paper has been supported by two research projects from the University of La Laguna, grant numbers 221 43/99 and 180221024.

References (15)

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