Tree-shaped Facility Research Articles

Combinatorial algorithms for the minmax robust median subtree problem on tree networks with interval vertex weights

The typical median subtree problem on trees is to locate a (continuous) tree-shaped facility (the leaves of this facility are not necessary vertices) with a specified length to minimize the total weighted distance of all vertices to the new facility. In this paper, each vertex weight can be any value within an interval and the total deviation of weights is limited within a threshold. We want to find a subtree that minimizes the median function in the worst-case scenario. This is the so-called minmax robust median subtree problem on trees. In order to solve the problem, we first consider the minmax robust 1-median problem on trees and solve the problem in [Formula: see text] time. Then, the nestedness property is coined, i.e., any optimal subtree must contain a robust 1-median in the tree. Based on this property, we develop a greedy algorithm that solves the corresponding problem in [Formula: see text] time.

Locating a discrete subtree of minimum variance on trees: New strategies to tackle a very hard problem

In this paper we study the problem of finding a discrete subtree of minimum variance with bounded size of a given tree. We show that the problem is NP-hard on very simple trees and it remains difficult even if we consider the Mean Absolute Deviation criterion on star graphs. Referring to the variance criterion, we first propose a reformulation of the problem as an Integer Quadratic Program that, however, does not have particular structural properties that may help in finding good lower bounds on the optimal value of the problem. Then, basing on recent results on conic optimization, we investigate alternative reformulations of the problem in order to obtain some lower bounds. In particular, we exploit a continuous linear, conic, reformulation of the problem and show that this is, in fact, an exact linear reformulation of the Integer Quadratic Program. Alternatively, by referring to the upper envelope approach, we show how to obtain a lower bound for the optimal objective function value of our discrete tree-shaped facility location problem in pseudo-polynomial time.

On the Location of a Constrained k-Tree Facility in a Tree Network with Unreliable Edges

Given a tree network T with n vertices where each edge has an independent operational probability, we are interested in finding the optimal location of a reliable service provider facility in a shape of subtree with exactly k leaves and with a diameter of at most l which maximizes the expected number of nodes that are reachable from the selected subtree by operational paths. Demand requests for service originate at perfectly reliable nodes. So, the major concern of this paper is to find a location of a reliable tree-shaped facility on the network in order to provide a maximum access to network services by ensuring the highest level of network connectivity between the demand nodes and the facility. An efficient algorithm for finding a reliable (k,l) – tree core of T is developed. The time complexity of the proposed algorithm is Olkn. Examples are provided to illustrate the performance of the proposed algorithm.

Half-integrality of node-capacitated multiflows and tree-shaped facility locations on trees

In this paper, we establish a novel duality relationship between node-capacitated multiflows and tree-shaped facility locations. We prove that the maximum value of a tree-distance-weighted maximum node-capacitated multiflow problem is equal to the minimum value of the problem of locating subtrees in a tree, and the maximum is attained by a half-integral multiflow. Utilizing this duality, we show that a half-integral optimal multiflow and an optimal location can be found in strongly polynomial time. These extend previously known results in the maximum free multiflow problems. We also show that the set of tree-distance weights is the only class having bounded fractionality in maximum node-capacitated multiflow problems.

Optimal Algorithms for the Path/Tree-Shaped Facility Location Problems in Trees

Extensive facility location models in networks deal with the location of special types of subgraphs such as paths or trees and can be considered as extensions of classical facility location models. In this paper we consider the problem of locating a path-shaped or tree-shaped (extensive) facility in trees, under the condition that existing facilities are already located. We introduce a parametric-pruning method to solve the conditional discrete/continuous extensive weighted 1-center location problems in trees in linear time. This improves the recent results of O(nlog n) by Tamir et al. (J. Algebra 56:50–75, [2005]).

Locating tree-shaped facilities using the ordered median objective

In this paper we consider the location of a tree-shaped facility S on a tree network, using the ordered median function of the weighted distances to represent the total transportation cost objective. This function unifies and generalizes the most common criteria used in location modeling, e.g., median and center. If there are n demand points at the nodes of the tree T=(V,E), this function is characterized by a sequence of reals, Λ=(λ1, . . . ,λn), satisfying λ1≥λ2≥...≥λn≥0. For a given subtree S let X(S)={x1, . . . ,xn} be the set of weighted distances of the n demand points from S. The value of the ordered median objective at S is obtained as follows: Sort the n elements in X(S) in nonincreasing order, and then compute the scalar product of the sorted list with the sequence Λ. Two models are discussed. In the tactical model, there is an explicit bound L on the length of the subtree, and the goal is to select a subtree of size L, which minimizes the above transportation cost function. In the strategic model the length of the subtree is variable, and the objective is to minimize the sum of the transportation cost and the length of the subtree. We consider both discrete and continuous versions of the tactical and the strategic models. We note that the discrete tactical problem is NP-hard, and we solve the continuous tactical problem in polynomial time using a Linear Programming (LP) approach. We also prove submodularity properties for the strategic problem. These properties allow us to solve the discrete strategic version in strongly polynomial time. Moreover the continuous version is also solved via LP. For the special case of the k-centrum objective we obtain improved algorithmic results using a Dynamic Programming (DP) algorithm and discretization and nestedness results.

Efficient Parallel Algorithms for Optimally Locating a Path and a Tree of a Specified Length in a Weighted Tree Network

In this paper, we propose efficient parallel algorithms on the EREW PRAM for optimally locating in a tree network a path-shaped facility and a tree-shaped facility of a specified length. Edges in the tree network have arbitrary positive lengths. Two optimization criteria are considered: minimum eccentricity and minimum distancesum. Let n be the number of vertices in the tree network. Our algorithm for finding a minimum eccentricity location of a path-shaped facility takes O(logn) time using O(n) work. Our algorithm for finding a minimum distancesum location of a path-shaped facility takes O(logn) time using O(n2) work. Both of our algorithms for finding the minimum eccentricity location and a minimum distancesum location of a tree-shaped facility take O(lognloglogn) time using O(n) work. In the sequential case, all the proposed algorithms are faster than those previously proposed by Minieka. Recently, Peng and Lo have proposed parallel algorithms for all the four problems considered in this paper. They assumed that each edge in the tree network is of length 1. Thus, as compared with their algorithms ours are more general. Besides, our algorithms for the problems of finding a minimum eccentricity location of a path-shaped facility, the minimum eccentricity location of a tree-shaped facility, and a minimum distancesum location of a tree-shaped facility are more efficient from the aspect of work. Their algorithms for these three problems use O(nlogn) work. Ours use O(n) work.

Fully polynomial approximation schemes for locating a tree-shaped facility: A generalization of the knapsack problem

Given an n-node tree T = ( V, E), we are concerned with the problem of selecting a subtree of a given length which optimizes the sum of weighted distances of the nodes from the selected subtree. This problem is NP-hard for both the minimization and the maximization versions since it generalizes the knapsack problem. We present fully polynomial approximation schemes which generate a (1 + ε)-approximation and a (1 − ε)-approximation for the minimization and maximization versions respectively, in O( n 2 g3 ) time.

On the location of a tree-shaped facility : Tae Ung Kim, Timothy J. Lowe, Arie Tamir and James E. Ward Networks Vol.28, 1996, pp. 167–175

On the location of a tree-shaped facility : Tae Ung Kim, Timothy J. Lowe, Arie Tamir and James E. Ward Networks Vol.28, 1996, pp. 167–175

On the location of a tree-shaped facility

This paper considers the problem of locating a central facility on a tree network. The central facility takes the form of a subtree of the network, and provides service to several demand points located at nodes of the network. Two types of costs are involved in evaluating a given facility. The setup cost represents the cost of establishing the facility and is taken to be directly proportional to the total length of the facility. The transportation cost is the cost associated with the travel of customers to the facility. The objective function is to select a tree-shaped facility that will minimize the sum of the setup cost and the total transportation cost. We introduce a general model and a simple {open_quotes}bottom-up{close_quotes} dynamic programming algorithm for its solution. We then focus on the important class of covering problems, where the transportation cost of a customer to the serving facility is zero if the distance between the two is below a certain threshhold, and is equal to some penalty term otherwise. We improve upon the performance of existing methods, and present algorithms with subquadratic complexity.

THE OPTIMAL LOCATION OF A STRUCTURED FACILITY IN A TREE NETWORK

Efficient parallel algorithms for finding an optimal path-shaped or tree-shaped facility with a specified size in a tree network are presented. Four kinds of optimization criteria are considered: minimizing/maximizing distancesum/eccentricity. There are eight cases when considering facility shapes and optimization criteria. Parallel algorithms for finding a minimum/maximum distancesum path were presented in [9]. The other six cases are studied in this paper. Two of these six cases can be solved optimally in linear Time x Processor complexity. For the problem of finding a maximum distancesum tree, the algorithm presented in this paper is the first polynomial time solution under a reasonable assumption.