The Kth Traveling Salesman Problem is Pseudopolynomial when TSP is polynomial

Given an undirected graph G = (V,E) with a weight function c ∈ R , and a positive integerK, the Kth Traveling Salesman Problem (KthTSP) is to findK Hamilton cyclesH1, H2, , ..., HK such that, for any Hamilton cycle H 6∈ {H1, H2, , ..., HK}, we have c(H) ≥ c(Hi), i = 1, 2, ...,K. This problem is NP-hard even for K fixed. We prove that KthTSP is pseudopolynomial when TSP is polynomial. 2010 Mathematics Subject Classification: Primary 90C27, Secondary 90C57.