Performance of a neural network method with set partitioning

Abstract

Computational experience with a neural network method for the set partitioning problem (i.e. minimize cx subject to Ax = 1, x j = 0 or 1, where A is a zero-one matrix, 1 is a column of ones, and c is a non-negative integral row) is presented. The approach maps the variables (subsets) of the problem onto equal numbers of neurons of a neural network model of Boltzmann machine-type operation. By mapping a neuron onto each subset rather than on each variable x ij = 0 or 1 depending on whether element i is in subset j or not (and then use the method of ‘penalization of constraints’, as was used by Hopfield for the travelling salesman problem [J.J. Hopfield, Proc. Natn. Acad. Sci. USA 81, 3088–3092 (1984); J.J. Hopfield and D.W. Tank, Biol. Cyber. 52, 141–152 (1985)]), the complexity of the neural network is considerably reduced. We propose an efficient encoding formalism which determines a good global behaviour of the model, around the valid partial or feasible solutions of the implemented problem. An analysis of the network's energy yields several interesting results that allow us to anticipate good performance. Simulations have been performed on sets of artificial data and real data encountered in the airline crew scheduling real application with very large sizes and very low density. Our numerical study, which extends to 8000 subsets, exhibits an impressive level of solution quality when the involved parameters are chosen with care. By exploring the central problem of scaling behaviour with respect to the size of the instances handled by our method, we gain some insight into the relationship between solutions quality, optimum values of the network tuning parameters, convergence speed and instances size.