Abstract :
The aim of this paper is to study experimentally divide and conquer strategies, used to implement parallel heuristics for the geometric Traveling Salesman Problem (TSP). Starting from Karpʹs results [R. M. Karp, Math. Ops Res. 2, 209–224 (1977)] we subdivide the set of cities into smaller sets, and we compute an optimal subtour for each subset. These subtours are then combined to obtain the tour for the entire problem. The analysis is restricted to problem instances where points are uniformly distributed in the unit square. For relatively small sets of cities we study the quality of the solution by comparing it with the optimal solution. In large problem-instances, statistical estimates of the optimal solutions are used for asymptotical analysis. We apply the same dissection strategy also to classical heuristics: the final solution, obtained by combining the approximate subsolutions, is compared with the average quality of the heuristic. Our main result is the estimate of the rate of convergence of the approximate solution to the optimal solution as a function of the number of dissection steps, of the criterion used for the plane division and of the quality of the subtours.
