A special case of the n-vertex traveling-salesman problem that can be solved in O( n) time

Abstract

The traveling-salesman problem, though in general NP-hard, possesses several special cases that can be solved in polynomial time. In particular, when the cost array C associated with an n-vertex traveling-salesman problem satisfies what are known as the Demidenko conditions, then a minimum-cost traveling-salesman tour can be computed in O( n 2) time using a simple dynamic-programming algorithm. In this paper, we identify a subset Λ of the set of all cost arrays satisfying the Demidenko conditions, such that for any C∈ Λ, the running time of the aforementioned dynamic-programming algorithm can be reduced to O( n). We obtain this speedup using recently developed techniques for on-line searching in Monge arrays.