Time complexity of the Analyst's Traveling Salesman algorithm

TL;DR

This work proves that the Analyst's Traveling Salesman Problem (ATSP) for finite sets in ℝ^N admits a polynomial-time algorithm, with a tight worst-case exponent of 3 and time bound $O(n^3)$. Building on Schul's and Badger–Naples–Vellis' ATSP constructions, the authors develop a graph-based, scale- and net-aware framework that enforces local flatness through flatness modules and flat pairs, guiding edge insertions while keeping the graph sparse. The method yields a connected graph G_m with V_m=V and O(n) edges, from which a tour is produced in $O(n^3)$ time; moreover, the algorithm provides computable Hölder-parameterizations for finite sets on rectifiable curves. The results connect geometric measure theory with computable curve construction, offering a concrete, efficient procedure for embedding finite point sets in finite-length curves and contributing to the broader understanding of ATSP in higher dimensions.

Abstract

The Analyst's Traveling Salesman Problem asks for conditions under which a (finite or infinite) subset of $\mathbb{R}^N$ is contained on a curve of finite length. We show that for finite sets, the algorithm constructed by Schul (2007)and Badger-Naples-Vellis (2019) that solves the Analyst's Traveling Salesman Problem has polynomial time complexity and we determine the sharp exponent.

Theorems & Definitions (16)

• Theorem 1.1
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 3.1
• proof
• Lemma 4.1
• proof
• Remark 4.2