A finite algorithm for the continuousp-center location problem on a graph (original) (raw)

A note on the p - center problem

Yugoslav Journal of Operations Research, 2011

The p-center problem is to locate p facilities in a network so as to minimize the longest distance between a demand point and its nearest facility. In this paper, we give a construction on a graph G which produces an infinite ascending chain 0 1 2 ... G G G G = ≤ ≤ ≤ of graphs containing G such that given any optimal solution X for the p-center problem on G , X is an optimal solution for the p-center problem on i G for any 1 i ≥ .

New Results on the Complexity of p-Centre Problems

SIAM Journal on Computing, 1983

An O(n log3 n) algorithm for the continuous p-center problem on a tree is presented. Following a sequence of previous algorithms, ours is the first one whose time bound in uniform in p and less than quadratic in n. We also present an O(n log2 n log log n) algorithm for a weighted discrete p-center problem.

On alternativep-center problems

Mathematical Methods of Operations Research, 1992

Let G = (V, E) be an undirected connected graph with positive edge lengths. The vertex p-center problem is to find the optimal location of p centers so that the maximum distance to a vertex from its nearest center is minimized, where the centers may be placed at the vertices. Kariv and Hakimi have shown that this problem is NP-hard. We will consider two modifications of this problem in which each center may be located in one of two predetermined vertices. We will show the NP-completeness of their recognition versions.

The p-center location problem in an area

Location Science, 1996

The p-center problem seeks the location of p facilities. Each demand point receives its service from the closest facility. The objective is to minimize the maximal distance for all demand points. In this paper, the p-center location problem for demand originating in an area is investigated. This problem is equivalent to covering every point in the area by p circles with the smallest possible radius. Heuristic procedures are proposed and upper bounds on the optimal solution in a square are given. Computational results for the special case of a square area are reported. Some cases such as p=9 centers in a square yield unexpected and interesting results.

The complete vertex p-center problem

The Complete Vertex p-Center Problem, 2020

The vertex p-center problem consists of locating p facilities among a set of M potential sites such that the maximum distance from any demand to its closest located facility is minimized. The complete vertex p-center problem solves the p-center problem for all p from 1 to the total number of sites, resulting in a multi-objective trade-off curve between the number of facilities and the service distance required to achieve full coverage. This trade-off provides a reference to planners and decision makers, enabling them to easily visualize the consequences of choosing different coverage design criteria for the given spatial configuration of the problem. We present two fast algorithms for solving the complete p-center problem: one using the classical formulation but trimming variables while still maintaining optimality and the other converting the problem to a location set covering problem and solving for all distances in the distance matrix. We also discuss scenarios where it makes sense to solve the problem via brute-force enumeration. All methods result in significant speedups, with the set covering method reducing computation times by many orders of magnitude.

On the Complexity of Some Common Geometric Location Problems

SIAM Journal on Computing, 1984

Given n demand points in the plane, the p-center problem is to find p supply points (anywhere in the plane) so as to minimize the maximum distance from a demo& point to its respective nearest supply point. The p-median problem is to minimize the sum of distances from demand points to their respective nearest supply points. We prove that the p-center and the p-media problems relative to both the Euclidean and the rectilinear metrics are NP-hard. In fact, we prove that it is NP-hard even to approximate the p-center problems sufficiently closely. The reductions are from 3-satisfiability.

An improved algorithm for the p-center problem on interval graphs with unit lengths

Computers & Operations Research, 2007

The p-center problem is to locate p facilities in a network of n demand points so as to minimize the longest distance between a demand point and its nearest facility. We consider this problem by modelling the network as an interval graph whose edges all have unit lengths. We present an O(n) time algorithm for the problem under the assumption that the endpoints of the intervals are sorted, which improves on the existing best algorithm for the problem that has a run time of O(pn).

Solving the constrained p-center problem using heuristic algorithms

Applied Soft Computing, 2011

The p-center problem is one of the location problems that have been studied in operations research and computational geometry. This paper describes a compatible discrete space version of the heuristic Voronoi diagram algorithm. Since the algorithm gets stuck in local optimums in some cases, we apply a number of changes in the body of the algorithm with regard to the geometry of the problem, in a way that it can reach the global optimum with a high probability. Finally, a comparison between the results of these two algorithms on several test problems and a real-world problem are presented.

The conditionalp-center problem in the plane

Naval Research Logistics, 1993

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. 0 1993 John Wiley & Sons, Inc.