Solving traveling salesman problem by real space renormalization technique

A real space renormalization technique is introduced which is used in the study of critical phenomenon, to obtain approximate solutions of traveling salesman problems. In the procedure the author prepares in advance the basic optimized paths for 2×2 discrete space and stores those as a data set. In the actual calculation, the considered space covering all cities is divided into square cells of the size 1/2×1/2, 1/4×1/4, 1/8×1/8..., step by step, and a representative point is settled at the center of each cell, if there are cities in the area of cell. Thus all of the positions and paths become discrete, finding a shorter tour of the TSP can be easily and quickly accomplished only by calling a corresponding optimized path from data set. Computer simulation was performed for the case of 532 USA cities data. On average, a 37% longer path for 532 cities than the optimal one was obtained. The shortest tour among 20 different initial conditions was 29% longer