For dense digraphs with minimum degree above n/2 and independent edge costs, a small modification of Karp's patching algorithm produces an ATSP tour whose cost is (1+o(1)) times the optimal assignment cost, with high probability.
Hoeffding, Probability inequalities for sums of bounded random variables, Journal of the American Statistical Association 58 (1963) 13-30
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Karp's patching algorithm on dense digraph
For dense digraphs with minimum degree above n/2 and independent edge costs, a small modification of Karp's patching algorithm produces an ATSP tour whose cost is (1+o(1)) times the optimal assignment cost, with high probability.