Physics – Condensed Matter – Statistical Mechanics
Scientific paper
2000-02-02
Eur. Phys. J. B 16, 677 (2000)
Physics
Condensed Matter
Statistical Mechanics
7 pages, 4 figures. Revised version with changes in text and figures (to be published in Euro. Phys. Jour. B)
Scientific paper
10.1007/PL00011064
If one places N cities randomly on a lattice of size L, we find that the normalized optimal travel distances per city in the Euclidean and Manhattan metrics vary monotonically with the city concentration p. We have studied such optimal tours for visiting all the cities using a branch and bound algorithm, giving exact optimized tours for small system sizes (N<100). Extrapolating the results for N tending to infinity, we find that the normalized optimal travel distances per city in the Euclidean and Manhattan metrics both equal unity for p=1, and they reduce to about 0.74 and 0.94, respectively, as p tends to zero. Although the problem is trivial for p=1, it certainly reduces to the standard TSP on continuum (NP-hard problem) for p tending to zero. We did not observe any irregular behaviour at any intermediate point. The crossover from the triviality to the NP-hard problem seems to occur at p=1.
Chakrabarti Bikas K.
Chakraborti Anirban
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