Modelling and solving central cycle problems with integer programming

Abstract

We consider the problem of identifying a central subgraph of a given simple connected graph. The case where the subgraph comprises a discrete set of vertices is well known. However, the concept of eccentricity can be extended to connected subgraphs such as: paths, trees and cycles. Methods have been reported which deal with the requirement that the subgraph is a path or a constrained tree. We extend this work to the case where the subgraph is required to be a cycle. We report on computational experience with integer programming models of the problems of identifying cycle centres, cycle medians and cycle centroids. The problems have applications in facilities location, particularly the location of emergency facilities, and service facilities. Scope and purpose This paper looks at ways of locating a circuit (cycle), such as an inner city rail link or a delivery or collection route, so that all customers wanting to use the link or be near to the delivery or collection route have minimal inconvenience in reaching it. The problem is looked at from a fairly abstract point of view and ways of modelling different types of cycles are considered. These cycles will represent sets of facilities located in a circular structure. Practical applications of the model are considered and discussed. Integer programming models are built of three different cycle problems—the cycle centre problem, the cycle median problem and the cycle centroid problem. The cycle centre problem corresponds to locating a cycle of facilities so the furthest distance travelled by any customer is minimised. The cycle median problem locates the cycle such that it minimises the total distance travelled by all customers and the cycle centroid problem essentially locates the cycle in order to minimise all distances travelled by customers. The models are then tested out on instances of networks that are generated to simulate realistic features of customer location and optimal solutions (optimal cycles) are sought and obtained. Computational experience is reported but it becomes clear that although the exact methods perform adequately on small to medium sized networks they cannot easily solve large instances of the models. For large instances approximate solutions would need to be the fallback position.