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.
Broder, How hard is it to marry at random, ? (On the approximation of the permanent), Proceedings of the 18th Annual ACM Symposium on Theory of Computing , ACM Press, 1986, 50–58
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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.