G-networks and minimum cost functions

Since Hopfield´s seminal work on energy functions for neural networks and their consequence for the approximate solution of optimization problems, much attention has been devoted to neural heuristics for combinatorial optimization. These heuristics are often very time consuming, because of the need for randomization or Monte Carlo simulation during the search for solutions. The class of G-networks have the nice property of being analytically solvable, product form queueing networks. Yet they have nonlinear properties (similar to those of Hopfield and other neural networks) which allow them to address similar issues such as learning and optimization. In this paper we first recall the basic queueing network model with positive and negative customers (G-network) and show that-in steady state-it minimizes a cost function which can be used for optimization problems. We illustrate this by the search for heuristic solutions to the minimum node covering problem (MCP) for graphs, which we then proceed to solve approximately using the G-network