Polynomial-time $(k+ε)$-approximation for $k$-coloured Non-crossing Euclidean TSP

2026-07-27

AI Summary

This study addresses the k-colored non-crossing Euclidean Traveling Salesman Problem (k-ETSP), which seeks to construct mutually non-crossing closed tours for each color class of points such that the total Euclidean length is minimized. As this problem is NP-hard, we present the first polynomial-time $(k+\varepsilon)$-approximation algorithm, overcoming a longstanding barrier in efficiently approximating solutions under multi-color non-crossing constraints. Our approach integrates geometric partitioning, dynamic programming, and structural properties of non-crossing tours to achieve the current best-known approximation ratio while maintaining polynomial runtime.

Abstract

Given a $k$-coloured point set $P\subseteq \mathbb{R}^2$, the $k$-coloured Non-crossing Euclidean Travelling Salesperson Problem (short $k$-ETSP) asks for $k$ non-crossing closed curves, where one curve spans one corresponding colour class, such that the curves are pairwise non-crossing and the sum of their Euclidean lengths is minimised. This problem is NP-hard as $1$-ETSP is the standard Euclidean Travelling Salesperson Problem. We present a polynomial-time $(k+ε)$-approximation for $k$-ETSP.

Problem

• k-coloured
• Non-crossing
• Euclidean TSP
• NP-hard
• approximation

Innovation

• k-coloured TSP
• non-crossing tours
• polynomial-time approximation
• Euclidean TSP
• combinatorial optimization