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)<β(n−1)! for any constant β>0 provided P≠NP. We prove that μ(n)<β(n−k)! for any fixed integer k⩾1 and constant β>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.