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You can find much of this out by trying

sage: g.traveling_salesman_problem??

to read the code. Note that in this case the function itself is in a .py (uncompiled, though in byte code I think) file. But the line

from sage.numerical.mip import MixedIntegerLinearProgram

indicates it uses MILP, and this is indeed a compiled file (as you can see by doing

sage: sage.numerical.mip.MixedIntegerLinearProgram?

where it indicates an .so file type.

However, in truth this calls an outside program (which you can set, e.g. CPLEX) to do the LP, so the compiled piece is less relevant than the fact that it uses a dedicated library to do the LP.