Is travelling salesman problem with integer weight NP-hard? I wonder if travelling salesman problem remains to be NP-hard with an additional constraint that the edge weight is integer.

The travelling salesman problem already has integer edge weights! For example, in Garey & Johnson, Computers and Intractability, the problem is defined as follows:

TRAVELLING SALESMAN
INSTANCE: A finite set $C=\{c_1,c_2,\ldots,c_m\}$ of "cities", a "distance" $d(c_i,c_j)\in Z^+$ for each pair of cities $c_i,c_j\in C$, and a bound $B\in Z^+$ (where $Z^+$ denotes the positive integers).
QUESTION: Is there a "tour" of all the cities in $C$ having total length no more than $B$?

In order for a problem to be in NP, it must be possible to verify a solution to the problem in time that's polynomial in the size of the problem description. For integer distances, it's obvious that this can be done. For fractional distances, it's not obvious: if you try to convert them all to a common denominator, how big might the denominator get? I think it can be done (the common denominator has size that's polynomial in the size of the problem description), but it's an awkward complication and it's easy to see why the problem is usually defined with integer distances.
