TSP on manifolds

TL;DR

This paper introduces a new PTAS for the TSP problem on bounded-curvature surfaces, extending previous algorithms to broader metric spaces through novel geometric embeddings.

Contribution

The paper defines a new class of surfaces based on Riemannian geometry and proves TSP admits a PTAS on these surfaces, generalizing prior geometric approximation algorithms.

Findings

TSP admits a PTAS on bounded-curvature surface spaces.

Every bounded doubling dimension space can be embedded into such a surface.

Every uniform metric space can be embedded into a bounded-curvature surface.

Abstract

In this paper, we present a new approach of creating PTAS to the TSP problems by defining a bounded-curvature surface embedded spaces. Using this definition we prove: - A bounded-curvature surface embedded spaces TSP admits to a PTAS. - Every bounded doubling dimension space can be embedded into a bounded-curvature surface. - Every uniform metric space can be embedded into a bounded-curvature surface. Thus, the algorithm generalizes arXiv:1112.0699 (and therefore [7] and [8] as well, w.r.t PTAS of TSP). But, the algorithm is much broader as uniform metric spaces aren't bounded doubling dimension spaces. It should be mentioned that our definition of a surface is derived from Riemannian geometry, but doesn't match it exactly. therefore, our definitions and basic geometry algorithm is given here in full. [7] Sanjeev Arora. 1998. Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. J. ACM 45, 5 (September 1998), 753-782. DOI=http://dx.doi.org/10.1145/290179.290180 [8] Joseph S. B. Mitchell. Guillotine subdivisions approximate polygonal subdivisions: A simple polynomial time approximation scheme for geometric TSP, k- MST, and related problems. SIAM J. Comput., 28(4):1298-1309, 1999.