Approximation and Hardness of Polychromatic TSP

TL;DR

This paper introduces the Polychromatic Traveling Salesman Problem (PCTSP), providing a polynomial-time approximation algorithm for metric cases and proving APX-hardness for Euclidean instances, highlighting its computational complexity.

Contribution

The paper defines PCTSP as a generalization of TSP and Bipartite TSP, offering the first approximation algorithm and hardness results for this problem.

Findings

Polynomial-time (3 - 2 * 10^{-36})-approximation for metric PCTSP.

Euclidean PCTSP is APX-hard in R^2, ruling out PTAS.

PCTSP generalizes classical TSP and Bipartite TSP.

Abstract

We introduce the Polychromatic Traveling Salesman Problem (PCTSP), where the input is an edge weighted graph whose vertices are partitioned into $k$ equal-sized color classes, and the goal is to find a minimum-length Hamiltonian cycle that visits the classes in a fixed cyclic order. This generalizes the Bipartite TSP (when $k = 2$) and the classical TSP (when $k = n$). We give a polynomial-time $(3 - 2 * 10^{-36})$-approximation algorithm for metric PCTSP. Complementing this, we show that Euclidean PCTSP is APX-hard even in $R^2$, ruling out the existence of a PTAS unless P = NP.