Efficient PTAS for the Maximum Traveling Salesman Problem in a Metric Space of Fixed Doubling Dimension

TL;DR

This paper introduces an efficient polynomial-time approximation scheme (PTAS) for the Max TSP in metric spaces with fixed doubling dimension, achieving near-optimal solutions in cubic time.

Contribution

It presents the first PTAS for Max TSP in arbitrary fixed doubling dimension metric spaces, extending geometric approximation techniques to more general metric settings.

Findings

Provides a cubic-time PTAS for Max TSP in fixed doubling dimension spaces.

Achieves asymptotically optimal solutions in fixed and sublogarithmic doubling dimensions.

Extends approximation algorithms beyond Euclidean spaces to general metric spaces.

Abstract

The maximum traveling salesman problem (Max TSP) consists of finding a Hamiltonian cycle with the maximum total weight of the edges in a given complete weighted graph. This problem is APX-hard in the general metric case but admits polynomial-time approximation schemes in the geometric setting, when the edge weights are induced by a vector norm in fixed-dimensional real space. We propose the first approximation scheme for Max TSP in an arbitrary metric space of fixed doubling dimension. The proposed algorithm implements an efficient PTAS which, for any fixed $\varepsilon\in(0,1)$, computes a $(1-\varepsilon)$-approximate solution of the problem in cubic time. Additionally, we suggest a cubic-time algorithm which finds asymptotically optimal solutions of the metric Max TSP in fixed and sublogarithmic doubling dimensions.