  If one places N cities on a continuum in an unit area, extensive numerical
results and their analysis (scaling, etc.) suggest that the best normalized
optimal travel distance becomes 0.72 for the Euclidean metric and 0.92 for the
Manhattan metric. The analytic bounds, we discuss here, give 0.5 and 0.92 as
the lower and upper bounds for the Euclidean metric, and 0.64 and 1.17 for the
Manhattan metric. When the cities are randomly placed on a lattice with
concentration p, we find that the normalized optimal travel distance vary
monotonically with p. For p=1, the values in both Euclidean and Manhattan
metric are 1, and as p tends to zero, the values are 0.72 and 0.92 in the
Euclidean and Manhattan metrics respectively.The problem is trivial for p=1 but
it reduces to the continuum TSP as p tends to zero. We do not get any irregular
behaviour at any intermediate point, e.g., the percolation point. The crossover
from the triviality to the NP- hard problem seems to occur at p<1.
