The nestedness property of the convex ordered median location problem on a tree

Abstract

This paper deals with the problem of locating an extensive facility of restricted length within a given tree network. Topologically, the selected extensive facility is a subtree. The nestedness property means that a solution of a problem with a shorter length constraint is part of a solution of the same problem with a longer length constraint. We prove the existence of a nestedness property for a common family of convex ordered median (COM) objective functions. We start with the proof of the nestedness property for a rooted tree problem, where the extended facility is a subtree of some tree network rooted at a specified node, and proceed to prove the nestedness property for the general location model on a tree.

Introduction

Generally, even in single resource constrained optimization problems, when the amount of the resource increases, say, from $t$ to $t^{\prime }$, the optimal solution vector $X\left(t^{\prime }\right)$, denoting the optimal “activity levels” for resource level $t^{\prime }$, is not necessarily greater than or equal to $X\left(t\right)$, the respective optimal solution for resource level $t$.

Monotonicity of the optimal set, in terms of the resource level $t$, is a very desirable property. Consider, for example, an optimization model where an initial budget level $t_0$ is allocated to a project, e.g., paving a segment of a dirt road, connecting a set of isolated small towns. Also, from past experience there is a good chance that some additional budget, say $s$, might be allocated at a later stage. The decision problem is whether to implement the optimal policy $X\left(t_0\right)$, or delay the implementation until the final budget, $t_0+s$, is hopefully revealed. Clearly, if the model possesses the nestedness (monotonicity) property, i.e., $X\left(t_0\right)\le X(t_0+s)$, implementation of $X\left(t_0\right)$ should be started immediately, even when the value of $s$ is not yet revealed. (In this model, $X\left(t\right)$ is the location of the optimal segment of length or budget $t$, along the dirt road.) This model is discussed in [1], for a family of objective functions, which depend on the weighted distances of the towns along the dirt road from the paved segment. This family is shown to possess the nestedness property.

We are aware of only few general optimization models that possess the nestedness property, i.e., there is no reason to start from scratch, and $X\left(t\right)$ can always be augmented to find $X\left(t^{\prime }\right)$. We give some examples.

The most well known example is maximizing a linear function over a polymatroidal system. The validity of the greedy algorithm proves the nestedness property for this model [2], [3]. Recursively, if an optimal solution $X\left(t\right)$ is already known for $t$ units of the resource, then $X(t+1)$ is obtained by adding one unit of maximum reward, such that the augmented solution satisfies the polymatroidal constraints. A related result holds in the case of the maximization of a concave and separable objective function over a polymatroidal system [4], [5]. A second example comes from directed two terminal series–parallel graphs. Both, the max-flow and the min-cost flow problems have the nestedness property, see [6], [7]. Again, recursively, if $X\left(t\right)$, the minimum cost flow of $t$ units from source to destination is known, $X(t+1)$ is computed by finding the least cost path in the residual series–parallel graph. A third example is the family of packing problems, defined by Greedy matrices. Such matrices are related to multiserver location problems defined on tree graphs [8].

More examples with the nestedness property have appeared in the location theory literature [9], [10]. In fact, our paper was motivated by these families of location problems defined on tree networks: Consider an undirected tree network $T=(V,E)$, where the nodes are the customers, and they are associated with positive weights. The edges have positive lengths and they are assumed to be rectifiable, so that we can refer to the continuum set of points along the edges of $T$. In fact, we can assume that $T$ is embedded in the plane, and the edges are represented by non-intersecting close segments of the respective edge lengths. $A\left(T\right)$ denotes the metric space induced by the positive edge lengths on this continuum set. The goal in tree location models is to locate an extensive server, which is modeled by a closed subtree of $A\left(T\right)$, (a compact and connected subset), of total length not exceeding a given real $t$. For a subtree $S$ with the total length denoted by $L\left(S\right)$, the objective is to minimize some isotone function $f\left(S\right)$ of the weighted distances of the customers from the extensive facility $S$. There are results in the literature, showing several objective functions that possess the nestedness property [1], [11], [12], [13], [14], [15]. The above location model is called the general tactical subtree location problem.

Our work extends the results in [1] from path graphs to trees. We show that the nestedness property holds for the above subtree location model with any convex ordered median function. The formal definition of this family of objectives, which has been used extensively in location theory over the last two decades, is given in the next section.

The paper is organized in four sections. In Section 2 we provide a formulation of the tactical extensive facility location problem and the nestedness property. The nestedness property for the rooted problem is proved in Section 3. Section 4 extends the proof for the general (unrooted) extensive facility location problem. The appendix (Appendix A) contains some lemmas used in the paper, without the proofs. Detailed proofs of the lemmas can be found in [16].