Upper bounds on ATSP neighborhood size

We consider the Asymmetric Traveling Salesman Problem (ATSP) and use the definition of neighborhood by Deineko and Woeginger (see Math. Programming 87 (2000) 519-542). Let µ(n) be the maximum cardinality of polynomial time searchable neighborhood for the ATSP on n vertices. Deineko and Woeginger conjectured that µ(n) 0 provided P ≠ NP. We prove that µ(n) 0 provided NP ⊈ P/poly, which (like P ≠ NP) is believed to be true. We also give upper bounds for the size of an ATSP neighborhood depending on its search time.

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