When is the Assignment Bound Tight for the Asymmetric Traveling Salesman Problem?

We consider the probabilistic relationship between the value of a random asymmetric traveling salesman problem ATSP (M) and the value of its assignment relaxation AP (M). We assume here that the costs are given by an n × n matrix M whose entries are independently and identically distributed. We focus on the relationship between Pr(ATSP (M) = AP (M)) and the probability pn that any particular entry is zero. If npn → ∞ with n then we prove that ATSP (M) = AP (M) with probability 1-o(1). This is shown to be best possible in the sense that if np(n) → c, c > 0 and constant, then Pr(ATSP (M) = AP (M)) < 1 − φ(c) for some positive function φ. Finally, if npn → 0 then Pr(ATSP (M) = AP (M)) → 0.