The Kth Traveling Salesman Problem is Pseudopolynomial when TSP is polynomial

Abstract

Given an undirected graph G = (V, E) with a weight function c ∈ R E , and a positive integer K, the Kth Traveling Salesman Problem (KthTSP) is to find K Hamilton cycles H 1 , H 2 , , ..., H K such that, for any Hamilton cycle H ∈ {H 1 , H 2 , , ..., H K }, we have c(H) ≥ c(H i ), i = 1, 2, ..., K. This problem is NP-hard even for K fixed. We prove that KthTSP is pseudopolynomial when TSP is polynomial.