GNU Scientific Library – Reference Manual: Traveling Salesman Problem

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#### 25.3.2 Traveling Salesman Problem

The TSP (Traveling Salesman Problem) is the classic combinatorial optimization problem. I have provided a very simple version of it, based on the coordinates of twelve cities in the southwestern United States. This should maybe be called the Flying Salesman Problem, since I am using the great-circle distance between cities, rather than the driving distance. Also: I assume the earth is a sphere, so I don’t use geoid distances.

The `gsl_siman_solve` routine finds a route which is 3490.62 Kilometers long; this is confirmed by an exhaustive search of all possible routes with the same initial city.

The full code can be found in `siman/siman_tsp.c`, but I include here some plots generated in the following way:

    $ ./siman_tsp > tsp.output
    $ grep -v "^#" tsp.output
     | awk '{print $1, $NF}'
     | graph -y 3300 6500 -W0 -X generation -Y distance
        -L "TSP - 12 southwest cities"
     | plot -Tps > 12-cities.eps
    $ grep initial_city_coord tsp.output
      | awk '{print $2, $3}'
      | graph -X "longitude (- means west)" -Y "latitude"
         -L "TSP - initial-order" -f 0.03 -S 1 0.1
      | plot -Tps > initial-route.eps
    $ grep final_city_coord tsp.output
      | awk '{print $2, $3}'
      | graph -X "longitude (- means west)" -Y "latitude"
         -L "TSP - final-order" -f 0.03 -S 1 0.1
      | plot -Tps > final-route.eps

This is the output showing the initial order of the cities; longitude is negative, since it is west and I want the plot to look like a map.

    # initial coordinates of cities (longitude and latitude)
    ###initial_city_coord: -105.95 35.68 Santa Fe
    ###initial_city_coord: -112.07 33.54 Phoenix
    ###initial_city_coord: -106.62 35.12 Albuquerque
    ###initial_city_coord: -103.2 34.41 Clovis
    ###initial_city_coord: -107.87 37.29 Durango
    ###initial_city_coord: -96.77 32.79 Dallas
    ###initial_city_coord: -105.92 35.77 Tesuque
    ###initial_city_coord: -107.84 35.15 Grants
    ###initial_city_coord: -106.28 35.89 Los Alamos
    ###initial_city_coord: -106.76 32.34 Las Cruces
    ###initial_city_coord: -108.58 37.35 Cortez
    ###initial_city_coord: -108.74 35.52 Gallup
    ###initial_city_coord: -105.95 35.68 Santa Fe

The optimal route turns out to be:

    # final coordinates of cities (longitude and latitude)
    ###final_city_coord: -105.95 35.68 Santa Fe
    ###final_city_coord: -103.2 34.41 Clovis
    ###final_city_coord: -96.77 32.79 Dallas
    ###final_city_coord: -106.76 32.34 Las Cruces
    ###final_city_coord: -112.07 33.54 Phoenix
    ###final_city_coord: -108.74 35.52 Gallup
    ###final_city_coord: -108.58 37.35 Cortez
    ###final_city_coord: -107.87 37.29 Durango
    ###final_city_coord: -107.84 35.15 Grants
    ###final_city_coord: -106.62 35.12 Albuquerque
    ###final_city_coord: -106.28 35.89 Los Alamos
    ###final_city_coord: -105.92 35.77 Tesuque
    ###final_city_coord: -105.95 35.68 Santa Fe

Here’s a plot of the cost function (energy) versus generation (point in the calculation at which a new temperature is set) for this problem:

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