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 k1 and constant β>0 provided NPP/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.