A PTAS for the Min-Max Euclidean Multiple TSP

TL;DR

This paper introduces a PTAS for the Euclidean min-max multiple TSP, providing near-optimal solutions efficiently by extending Arora's Euclidean TSP approach with a novel rounding process.

Contribution

It develops a PTAS for the min-max multiple TSP in Euclidean space, extending Arora's dynamic programming method with a new balancing technique.

Findings

Achieves a (1 + ε)-approximation for the problem.

Provides an algorithm with runtime polynomial in n for fixed ε.

Introduces a rounding process to balance tour lengths.

Abstract

We present a polynomial-time approximation scheme (PTAS) for the min-max multiple TSP problem in Euclidean space, where multiple traveling salesmen are tasked with visiting a set of $n$ points and the objective is to minimize the maximum tour length. For an arbitrary $\varepsilon > 0$, our PTAS achieves a $(1 + \varepsilon)$-approximation in time $O \big(n ((1/\varepsilon) \log (n/\varepsilon))^{O(1/\varepsilon)} \big)$. Our approach extends Sanjeev Arora's dynamic-programming (DP) PTAS for the Euclidean TSP (https://doi.org/10.1145/290179.290180). Our algorithm introduces a rounding process to balance the allocation of path lengths among the multiple salesman. We analyze the accumulation of error in the DP to prove that the solution is a $(1 + \varepsilon)$-approximation.