Solution of the Traveling-Salesman Problem

  • Berndt Müller
  • Joachim Reinhardt
Part of the Physics of Neural Networks book series (NEURAL NETWORKS)


As we know from the spin-glass analogy neural networks can model a complicated ‘energy landscape’ and hence can be employed for solving optimization problems by searching for minima in this energy surface. In Sect. 10.3.1 we have discussed an approach for solving the paragon combinatorial optimization task, the traveling salesman problem (TSP), using a neural network. Hopfield and Tank [Ho85] have mapped the N-city TSP onto a network with N × N formal neurons n iα. These are assumed to have a graded response with a continuous output value in the range 0 ... 1 determined by the local field u iα through the Fermi function
$$ {n_{i\alpha }}\,\, = \,\,\frac{1}{{1\,\, + \,\,{e^{ - 2{u_{i\alpha }}/T}}}} $$
with formal temperature T. The neurons are arranged in a square array, the row i denoting the number of the city and the column α denoting the station of the tour on which this city is visited. A valid tour is therefore characterized by an activation pattern with exactly N neurons ‘firing’ and N(N − 1) ‘quiescent’. There must be exactly one entry of 1 in each row and each column of the neural matrix n iα.


Travel Salesman Problem Travel Salesman Problem Spin Glass Valid Solution Spin Variable 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1990

Authors and Affiliations

  • Berndt Müller
    • 1
  • Joachim Reinhardt
    • 2
  1. 1.Department of PhysicsDuke UniversityDurhamUSA
  2. 2.Institut für Theoretische PhysikJ.-W.-Goethe-UniversitätFrankfurt 1Fed. Rep. of Germany

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