P-center Problem Research Articles

Center problems with pos/neg weights on trees

In a network with positive and negative vertex weights the pos/neg 1-center problem asks to minimize a linear combination of the maximum weighted distances of the center to the vertices with positive weights and to the vertices with negative weights, respectively. We show that in a network with n vertices and m edges the pos/neg 1-center problem can be solved in O( mnlog n) time. In trees a better complexity can be achieved. In the case of a path or of a star graph this problem can be solved in linear time. Further this problem is studied for a cactus with vertex weights 1 and −1. Moreover, an algorithm for the discrete anti- p-center problem on a tree with the improved time complexity O( nlog 2 n) is given. Finally, the pos/neg discrete p-center on a tree is treated and solved by an algorithm of time complexity O( n 2log n).

Parallel NC‐algorithms for multifacility location problems with mutual communication and their applications

AbstractThe generic problem studied is to locate p distinguishable facilities on a tree to satisfy upper‐bound constraints on distances between pairs of facilities, given that each facility must be located within its own feasible region, which is defined as a subtree of the tree. We present a parallel location scheme (PLS) for solving the problem that can be implemented as an NC‐algorithm. We also introduce parallel NC‐algorithms based on the PLS for the minimax versions of the problem, including the distance‐constrained p‐center problem with mutual communication. Combining the PLS and the improved Megiddo's parametric technique, we develop strongly polynomial serial algorithms for the minimax problems; the algorithms have the best complexities currently available in the literature. Efficient parallel algorithms are given for obtaining optimal regions of the facilities. © 2002 Wiley Periodicals, Inc.

Efficient algorithms for centers and medians in interval and circular‐arc graphs

AbstractThe p‐center problem is to locate p facilities on a network so as to minimize the largest distance from a demand point to its nearest facility. The p‐median problem is to locate p facilities on a network so as to minimize the average distance from a demand point to its closest facility. We consider these problems when the network can be modeled by an interval or circular‐arc graph whose edges have unit lengths. We provide, given the interval model of an n vertex interval graph, an O(n) time algorithm for the 1‐median problem on the interval graph. We also show how to solve the p‐median problem, for arbitrary p, on an interval graph in O(pn log n) time and on a circular‐arc graph in O(pn2 log n) time. We introduce a spring representation of the objective function and show how to solve the p‐center problem on a circular‐arc graph in O(pn) time, assuming that the arc endpoints are sorted. © 2002 Wiley Periodicals, Inc.

On Minimizing the Cost of Location Management in Mobile Environments

A mobile host typically has a home agent that maintains a registry of its current location. This registry is normally updated every time a host changes its current network. The update cost could be reduced using a two-tier update process in which a registry is updated using special agents, called proxy agents. We study the problem of selecting proxy agents to minimize the cost of search associated with this two-tier update approach. We show that the problem can be formulated as p-center or p-median finding problems. We focus on the p-center formulation. Due to the intractability of the problem, we introduce a distributed strategy to solve the general problem and show that it yields an approximate solution for arbitrary networks. We present an implementation of the distributed strategy that produces an optimal solution for ring networks. We prove that the optimal solution for rings is fault tolerant and resilient to topology changes.

The k-centrum multi-facility location problem

The most common problems studied in network location theory are the p-center and the p-median problems. In the p-center problem the objective is to locate p service facilities to minimize the maximum of the service distances of the n customers to their respective nearest service facility, and in the p-median model the objective is to minimize the sum of these n service distances. (A customer is served only by the closest facility.) We study the p-facility k-centrum model that generalizes and unifies the above problems. The objective of this unifying model is to minimize the sum of the k largest service distances. The p-center and the p-median problems correspond to the cases where k=1 and n, respectively. We present polynomial time algorithms for solving the p-facility k-centrum problem on path and tree graphs. These algorithms can be combined with the general approximation algorithms of Bartal (Proceedings of the 30th Annual ACM Symposium on Theory of Computing, 1998, pp. 161–168) and Charikar et al. (Proceedings of the 30th Annual ACM Symposium on Theory of Computing, 1998, pp. 114–123) to obtain an O( log n log log n) approximation for a p-facility k-centrum problem defined on a general network.

A p-center grid-positioning aggregation procedure

We present an algorithm for aggregating p-center demand points such that the resulting aggregate set imposes a grid structure on the plane consisting of identical “diamonds” of user-specified dimensions. The grid is positioned so that a well-defined upper bound on the objective function error is smallest. Scope and purpose A model often proposed for locating emergency service facilities is the p-center model. Its solution recommends locations for such facilities to minimize the maximum time (or distance) to respond to an emergency at some demand point. In an urban context, there could be millions of possible emergency demand points. Thus, it is common to aggregate demand points. The aggregation creates error. For a certain type of p-center problem, we present an aggregation procedure which can also be used to predict the maximum error, thus allowing the error to be kept within acceptable limits.

A polynomial algorithm for thep-centdian problem on a tree

The most common problems studied in network location theory are the p-median and the p-center models. The p-median problem on a network is concerned with the location of p points (medians) on the network, such that the total (weighted) distance of all the nodes to their respective nearest points is minimized. The p-center problem is concerned with the location of p-points (centers) on the network, such that the maximum (weighted) distance of all the nodes to their respective nearest points is minimized. To capture more real-world problems and obtain a good way to trade-off minisum (efficiency) and minimax (equity) approaches, Halpern introduced the centdian model, where the objective is to minimize a convex combination of the objective functions of the center and the median problems. In this paper, we studied the p-centdian problem on tree networks and present the first polynomial time algorithm for this problem. © 1998 John Wiley & Sons, Inc. Networks 32:255–262, 1998

A spanning tree approach to the absolute p-center problem

We consider the absolute p-center problem on a general network and propose a spanning tree approach which is motivated by the fact that the problem is NP-hard on general networks but solvable in polynomial time on trees. We first prove that every connected network possesses a spanning tree whose p-center solution is also a solution for the network under consideration. Then we propose two classes of spanning trees that are shortest path trees rooted at certain points of the network. We give an experimental study, based on 1440 instances, to test how often these classes of trees include an optimizing tree. We report our computational results on the performance of both types of trees.

Locating centers in a dynamically changing network, and related problems

In a dynamically changing network, the costs or distances between locations are changing in each discrete time period. We consider the location of emergency facilities that must minimize the maximum distance to any customer on the network across all time periods. We call the problem of locating p centers over k underlying networks corresponding to k periods the k-Network p-Center problem. The problem is considered when, in each period, the network satisfies the triangle inequality. In this paper, we provide a polynomial time 3-approximation algorithm for Δ k-Network p-Center for the case k=2. We discuss generalizations inspired by this problem to other optimization problems with multiple underlying networks and the objective of finding a single solution that varies as little as possible from the optimum for each network. The additional combinatorial problems discussed include: sorting; perfect matching; shortest path; minimum spanning tree; and minimum cut. All are shown to be NP-hard for k⩾2.

Location problems with grouped structure of demand: Complexity and algorithms

We study generalizations of classical multifacility location problems, where customers' demand has a hierarchial structure, i.e., the set of local customers is partitioned into categories (global customers), each having its own requirements for quality of service. For the case of identical facilities, we prove that the categorized coverage, covering, p-center, and p-median problems are strongly NP-hard on trees, in contrast with their classical noncategorized versions (which are polynomially solvable on trees). Some of the problems are shown to be NP-hard even on paths. For the case of dist nguishable facilities, we provide polynomial and strongly polynomial algorithms for the categorized covering, multicenter, and multimedian problems with mutual communication on a tree.

Minimax regret p-center location on a network with demand uncertainty

We consider the weighted p-center problem on a transportation network with uncertain weights of nodes. Specifically, for each node, an interval estimate of its weight is known. The objective is to find the ‘minimax regret’ solution i.e. to minimize the worst-case loss in the objective function that may occur because a decision is made without knowing which state of nature will take place. We discuss properties of the problem and show that the problem can be solved by means of solving ( n + 1) regular weighted p-center problems. This leads to polynomial algorithms for the cases where the regular weighted p-center problem can be solved in polynomial time, e.g. for the case of a tree network, and for the case of a general network with p = 1.

Generalized p-Center problems: Complexity results and approximation algorithms

In an earlier paper, two alternative p-Center problems, where the centers serving costumers must be chosen so that exactly one node from each of p prespecified disjoint pairs of nodes is selected, were shown to be NP-complete. This paper considers a generalized version of these problems, in which the nodes from which the p servers are to be selected are partitioned into k sets and the number of servers selected from each set must be within a prespecified range. We refer to these problems as the ‘Set’ p-Center problems. We establish that the triangle inequality (Δ-inequality) versions of these problems, in which the edge weights are assumed to satisfy the triangle inequality, are also NP-complete. We also provide a polynomial time approximation algorithm for the two Δ-inequality Set p-Center problems that is optimal for one of the problems in the sense that no algorithm with polynomial running time can provide a better constant factor performance guarantee, unless P = NP. For the special case ‘alternative’ p-Center problems, which we refer to as the ‘Pair’ p-Center problems, we extend the previous results in several ways. For example, the results mentioned above for the Set p-Center problems also apply to the Pair p-Center problems. Furthermore, we establish and exploit a correspondence between satisfiability and the dominating set type of problems that naturally arise when considering the decision versions of the Pair p-Center problems.

On a generalization of the p-Center Problem

We study a generalization of the p-Center Problem, which we call the α- Neighbor p-Center Problem ( p- Center (α) ). Given a complete edge-weighted network, the goal is to minimize the maximum distance of a client to its α nearest neighbors in the set of p centers. We show that in general finding a O(2 poly(¦V¦))- approximation for p- Center (α) is NP-hard, where ¦V¦ denotes the number of nodes in the network. If the distances are required to satisfy the triangle inequality, there can be no polynomial time approximation algorithm with a (2 − ε) performance guarantee for any fixed ε > 0 and any fixed α ⩽ p, unless P = NP. For this case, we present a simple yet efficient algorithm that provides a 4-approximation for α ⩾ 2. If α = 1, our algorithm basically falls back to the algorithm presented in [2] and has a relative performance guarantee of 2.

Polynomial algorithm for the equal capacity p-center problem on trees

Polynomial algorithm for the equal capacity p-center problem on trees

Technical Note—A Polynomial Algorithm for the Equal Capacity p-Center Problem on Trees

The uncapacitated p-center problem has been shown to be NP-Hard for the case of a general network, yet polynomial algorithms exist for the case of a tree network. We extend the results for trees to include the case where each center can serve a limited number of customers and show that the capacitated p-center problem on trees can be solved in polynomial time when the capacities are identical. The algorithm consists of solving a capacitated covering problem and then using search routines to find the optimal domination radius. We show that the addition of center capacities to either the vertex center problem or the absolute center problem requires a multiplicative factor of O(n) more work, relative to that required for the uncapacitated problems, to solve the related covering problems (n is the number of vertices in the tree).

Location-Allocation Models and their Development

Location-allocation modelling studies, primarily applied to facility Location problems, have rapidly increased and extensively spanned a wide range of disciplines in the past three decades. In Japan, facility location models have attracted the attention of planning-oriented researchers from architecture, city engineering, operations research, and of a few geographers. This paper has three aims: first, to introduce location-allocation models to Japanese geographers; second, to review the development of new literature on representing decision-making behavior, uncertainty in the environment and the multi-objective formulation; and, third, to identify the tasks of models in chiefly geographical research.In general, location-allocation models are classified into three problems: the p-median problem which minimizes the aggregate distance or time from demand point i to facility j; the p-center problem which minimizes the maximum distance from i to j; the covering problem where coverage is required (the set covering problem) or optimized (the maximal covering location problem). These models on a network are formulated as a linear programming or a mixed-integer programming. Due to the computational complexity of these problems, several heuristic solution methods have been developed and demand data aggregation methods to eliminate errors in the aggregation process have been proposed.With improved solution methods and increasing computational speed, location-allocation models have been adapted to complex and realistic location problems. A variety of facility problems and developed models are summarized in the following categories: First, considering the characteristics of a facility, models are diversely formulated by the type of facility, the existence of constraint on capacity and the hierarchy of the facility system. Second are the facilities-location methods, including free location, adding locations to an existing network, or restructuring of networks (Hodgart, 1978), which are linked to dynamic locational problems. The above options are mainly concerned with “location”and selected by a single-location decision-maker. However, the third“allocation” option is controlled by two decision-makers, namely, the locator and allocator (Leonardi, 1981). Various types of decision-making behavior will be represented by several allocation patterns. Fig. 3 shows the classification of allocation systems that contains nearest-neighbor allocation, probabilistic allocation, intermediate trips, combined and round trips, routing trips and the hub-facility problem. Later, realistic models reflecting an uncertain environment are presented. Uncertainties arise from the stochastic nature of demand, travel time, service provision and the competitive environment in the private sector. Finally, it is demonstrated that multi-objective location-allocation models solve the complexity of decision-making faced in realistic facility location planning.Most applications of location-allocation models typically placed emphasis on facility location planning. Location-allocation models will provide, if we geographers as location analysts seek another way to use models, valuable tools by wich well-defined location theories, especially central place theory, may be formulated as operational and normative models. Accordingly I propose two tasks for location-allocation models as follows:1) use of location-allocation models as a decision support system for facility location planning;2) use of location-allocation models as a spatial analysis tool. The latter task can be adapted to geographers interests and demonstrate spatial analysis including theoretical insights in understanding real-world spatial organization. In addition, linked with geographic information systems (GIS), location-allocation models could create more interactive tools.

The conditionalp-center problem in the plane

An algorithm is given for the conditional p-center problem, namely, the optimal location of one or more additional facilities in a region with given demand points and one or more preexisting facilities. The solution dealt with here involves the minimax criterion and Euclidean distances in two-dimensional space. The method used is a generalization to the present conditional case of a relaxation method previously developed for the unconditional p-center problems. Interestingly, its worst-case complexity is identical to that of the unconditional version, and in practice, the conditional algorithm is more efficient. Some test problems with up to 200 demand points have been solved. © 1993 John Wiley & Sons, Inc.

Discrete Location Theory

Ingredients of Locational Analysis (J. Krarup & P. Pruzan). The p-Median Problem and Generalizations (P. Mirchandani). The Uncapacitated Facility Location Problem (G. Cornuejols, et al.). Multiperiod Capacitated Location Models (S. Jacobsen). Decomposition Methods for Facility Location Problems (T. Magnanti & R. Wong). Covering Problems (A. Kolen & A. Tamir). p-Center Problems (G. Handler). Duality: Covering and Constraining p-Center Problems on Trees (B. Tansel, et al.). Locations with Spatial Interactions: The Quadratic Assignment Problem (R. Burkard). Locations with Spatial Interactions: Competitive Locations and Games (S. Hakimi). Equilibrium Analysis for Voting and Competitive Location Problems (P. Hansen, et al.). Location of Mobile Units in a Stochastic Environment (O. Berman, et al.). Index.

Technical Note—Conditional Location Problems on Networks

In this paper, we describe an algorithm to solve conditional location problems (such as the conditional p-median problem or the conditional p-center problem) on networks, where demand points are served by the closest facility whether existing or new. This algorithm requires the one-time solution of a (p + 1)-unconditional corresponding location problem using an appropriate shortest distance matrix.

Polynomial algorithms for restricted Euclidean p-centre problems

Given n demand points in the plane, the p-centre problem is to locate p supply points so as to minimize the maximum distance from a demand point to its nearest supply point. Megiddo and Supowit have recently shown that not only is this problem NP-hard, but even finding a close approximate solution to the problem is NP-hard. In this paper we present a polynomial time algorithm for the Euclidean p-centre problem when the demand points are restricted to lie on a fixed number of parallel lines.