New formulations for the Kissing Number Problem

Authors:
Sergei Kucherenko;Pietro Belotti;Leo Liberti;Nelson Maculan
Affiliations:
CPSE, Imperial College, SW7 2AZ London, UK;Tepper School of Business, Carnegie Mellon University, Pittsburgh PA 15213, USA;LIX, ícole Polytechnique, Palaiseau F-91128, France;COPPE, Universidade Federal do Rio de Janeiro, P.O. Box 68511, 21941-972 Rio de Janeiro, Brazil
Venue:
Discrete Applied Mathematics
Year:
2007

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Abstract

Determining the maximum number of D-dimensional spheres of radius r that can be adjacent to a central sphere of radius r is known as the Kissing Number Problem (KNP). The problem has been solved for two, three and very recently for four dimensions. We present two nonlinear (nonconvex) mathematical programming models for the solution of the KNP. We solve the problem by using two stochastic global optimization methods: a Multi Level Single Linkage algorithm and a Variable Neighbourhood Search. We obtain numerical results for two, three and four dimensions.