Approximation and Hardness of Polychromatic TSP

2025-07-07

AI Summary

This paper studies the Palette-Constrained Traveling Salesman Problem (PCTSP): given a metric graph whose vertices are partitioned into $k$ color classes of equal size, find the shortest Hamiltonian cycle that visits vertices of each color class in a fixed cyclic order. PCTSP unifies and generalizes both the classical TSP and the bipartite TSP. We formally introduce the PCTSP model and present the first polynomial-time approximation algorithm with approximation ratio strictly better than 3—specifically, $3 - 2 imes 10^{-36}$. Furthermore, we prove that PCTSP is APX-hard in two-dimensional Euclidean space, ruling out the existence of a PTAS unless P = NP. Our technical approach integrates degree-constrained subgraph construction, matching techniques, and graph traversal strategies, with rigorous analysis conducted in both general metric and Euclidean settings.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Generalizes Bipartite and classical TSP with colored vertices
• Seeks minimum-length cycle visiting color classes in order
• Proves approximation algorithm and APX-hardness for Euclidean PCTSP

Innovation

• Polychromatic TSP generalizes Bipartite and classical TSP
• Polynomial-time approximation algorithm for metric PCTSP
• Euclidean PCTSP is APX-hard in R^2