# The Kth Traveling Salesman Problem is Pseudopolynomial when TSP is polynomial

## TL;DR

This paper demonstrates that the Kth Traveling Salesman Problem becomes pseudopolynomial solvable when the classical TSP can be solved in polynomial time, highlighting a nuanced complexity relationship.

## Contribution

It establishes that KthTSP is pseudopolynomial under the condition that TSP is polynomial, providing new insights into the problem's complexity.

## Key findings

- KthTSP is NP-hard for fixed K.
- KthTSP is pseudopolynomial if TSP is polynomial.
- The result links TSP complexity to the KthTSP variant.

## Abstract

Given an undirected graph $G=(V, E)$ with a weight function $c\in R^E$, 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 $H\not \in \{H_1, H_2, , ..., H_K \}$, we have $c(H)\geq c(H_i), i=1, 2, ..., K$. This problem is NP-hard even for $K$ fixed. We prove that KthTSP is pseudopolynomial when TSP is polynomial.