Colored Points Traveling Salesman Problem
Saeed Asaeedi
TL;DR
This work defines Colored Points TSP, a color-constrained variant of the Traveling Salesman Problem where a set of $n$ points partitioned into $k$ colors must be visited by one representative per color. It establishes NP-hardness and presents two algorithms: an exact brute-force method with runtime $O(x_1 x_2 \cdots x_k)$ and a polynomial-time $(2\pi r/3)$-approximation that leverages the minimum color-spanning circle of radius $r$ and onion peeling to bound the perimeter. The authors implement both approaches in Matlab and provide numerical results on random datasets, showing that the exact method is feasible only for small instances while the approximation scales to larger ones with practical performance. The work contributes a formal definition, hardness results, and concrete algorithms with provable approximation guarantees, offering a tractable tool for problems requiring visiting exactly one representative from each color class across applications such as logistics and inspection.
Abstract
The Colored Points Traveling Salesman Problem (Colored Points TSP) is introduced in this work as a novel variation of the traditional Traveling Salesman Problem (TSP) in which the set of points is partitioned into multiple classes, each of which is represented by a distinct color (or label). The goal is to find a minimum cost cycle $C$ that visits all the colors and only makes each one appears once. This issue has various applications in the fields of transportation, goods distribution network, postal network, inspection, insurance, banking, etc. By reducing the traditional TSP to it, we can demonstrate that Colored Points TSP is NP-hard. Here, we offer a $\frac{2πr}{3}$-approximation algorithm to solve this issue, where $r$ denotes the radius of the points' smallest color-spanning circle. The algorithm has been implemented, executed on random datasets, and compared against the brute force method.