N-Oscillator Neural Network Based Efficient Cost Function For N-City Traveling Salesman Problem

Neural Networks have long been a mainstream technique to solve optimization problems. A classic example is the Travelling Salesman Problem (TSP) which NP-hard. Using a Hopfield-Tank representation, an n-city problem is mapped to a cost function of n(2) interacting neural units. Stochastic gradient descent helps achieve the global minima Due to the nature of the TSP problem, the cost function has to penalize invalid sub-routes (non-Hamiltonian cycles) and minimize the travel cost simultaneously. In addition, there is a starting point and travel direction associated '2n' degeneracy. Previously, a cellular neuronal approach was proposed where the neural units were replaced with oscillators. The phase relations determined the output solution. Multiphase clusters of these oscillators solved the degeneracy issue. This paper proposes an n-oscillator cost function for an n-city TSP. Since a group of single frequency oscillator phases are naturally ordered and circular in a system, the proposed method exploits the true potential of oscillator nodes. The sub-routes and degeneracy are eliminated by design in addition to massively increasing the scaling potential (n vs. n(2)). It was also found that the proposed n mapping can converge to the optimum tour much faster (about 100 times for a 5-city problem) than for n(2) mapping. Our approach projects hardware efficiency in terms of area footprint, computation time and energy. With coupled single device-based compact nanoscale oscillator systems becoming increasingly viable in hardware, efficient cost function mappings of hard problems using oscillator phases, as shown here, is critical to solving large graphical optimization problems.