[Paper Review] The Geometric Maximum Traveling Salesman Problem

Alexander Barvinok, Sándor P. Fekete+4 more|ArXiv.org|Apr 10, 2002

Computational Geometry and Mesh Generation20 references4 citations

TL;DR

This paper presents polynomial-time algorithms for the geometric Maximum Traveling Salesman Problem (MaxTSP) under polyhedral norms in fixed-dimensional Euclidean space. It shows that for any polyhedral norm with $ f $ facets, the MaxTSP can be solved in $ O(n^{f-2} ext{log}n) $ time, with an optimal $ O(n) $ algorithm for 2D cases like the rectilinear and sup norms, while proving NP-hardness for Euclidean distances in $ \mathbb{R}^d $ when $ d \geq 3 $.

ABSTRACT

We consider the traveling salesman problem when the cities are points in R^d for some fixed d and distances are computed according to geometric distances, determined by some norm. We show that for any polyhedral norm, the problem of finding a tour of maximum length can be solved in polynomial time. If arithmetic operations are assumed to take unit time, our algorithms run in time O(n^{f-2} log n), where f is the number of facets of the polyhedron determining the polyhedral norm. Thus for example we have O(n^2 log n) algorithms for the cases of points in the plane under the Rectilinear and Sup norms. This is in contrast to the fact that finding a minimum length tour in each case is NP-hard. Our approach can be extended to the more general case of quasi-norms with not necessarily symmetric unit ball, where we get a complexity of O(n^{2f-2} log n). For the special case of two-dimensional metrics with f=4 (which includes the Rectilinear and Sup norms), we present a simple algorithm with O(n) running time. The algorithm does not use any indirect addressing, so its running time remains valid even in comparison based models in which sorting requires Omega(n \log n) time. The basic mechanism of the algorithm provides some intuition on why polyhedral norms allow fast algorithms. Complementing the results on simplicity for polyhedral norms, we prove that for the case of Euclidean distances in R^d for d>2, the Maximum TSP is NP-hard. This sheds new light on the well-studied difficulties of Euclidean distances.

Motivation & Objective

• Address the computational complexity of the Maximum TSP under geometric distance functions in fixed-dimensional space.
• Identify which geometric norms allow polynomial-time solutions, contrasting with known NP-hardness of Minimum TSP under the same norms.
• Develop efficient algorithms for polyhedral norms by exploiting their combinatorial and geometric structure.
• Establish a sharp dichotomy between tractable polyhedral norms and intractable Euclidean norms in higher dimensions.
• Provide a theoretical foundation for understanding why certain distance functions yield efficient algorithms despite the general NP-hardness of TSP.

Proposed method

• Model polyhedral norms using a centrally symmetric polyhedron as the unit ball, with $ f $ facets, and express distances via the maximum of absolute dot products with facet normal vectors.
• Reduce the MaxTSP to a variant called the Tunneling TSP, leveraging the structure of the norm's dual representation.
• Use geometric sorting and convex hull techniques to identify optimal edge sets that maximize total tour length under the norm.
• Design a linear-time algorithm for 2D polyhedral norms with four facets (e.g., rectilinear and sup norms), avoiding indirect addressing to maintain $ O(n) $ complexity in comparison-based models.
• Prove NP-hardness for Euclidean MaxTSP in $ \mathbb{R}^d $, $ d \geq 3 $, via a reduction from Hamiltonian cycle using a carefully constructed point set with $ L_p $-like distances.
• Extend results to quasi-norms (non-symmetric unit balls), yielding $ O(n^{2f-2}\text{log}n) $ time complexity, and analyze structural differences in the solution space.

Experimental results

Research questions

• RQ1 Can the Maximum TSP be solved in polynomial time for geometric instances under polyhedral norms in fixed-dimensional space?
• RQ2 What is the computational complexity of the MaxTSP under Euclidean distances in $ \mathbb{R}^d $ for $ d \geq 3 $?
• RQ3 How does the number of facets of the unit ball influence the time complexity of solving the MaxTSP under a polyhedral norm?
• RQ4 Is there a linear-time algorithm for the MaxTSP in 2D under norms with four facets, such as the rectilinear and sup norms?
• RQ5 Can the structural properties of polyhedral norms be leveraged to design algorithms that avoid the $ \Omega(n\text{log}n) $ lower bound of comparison-based sorting?

Key findings

• For any fixed $ d $, the Maximum TSP under a polyhedral norm with $ f $ facets can be solved in $ O(n^{f-2}\text{log}n) $ time, demonstrating polynomial-time solvability.
• An optimal $ O(n) $-time algorithm exists for the 2D MaxTSP under 4-facet polyhedral norms, such as the rectilinear and sup norms, even in comparison-based models.
• The MaxTSP is NP-hard for Euclidean distances in $ \mathbb{R}^d $ when $ d \geq 3 $, establishing a sharp contrast with the tractable polyhedral cases.
• Quasi-norms with $ f $ facets yield an $ O(n^{2f-2}\text{log}n) $-time algorithm, extending the approach to non-symmetric distance functions.
• The MaxTSP under $ L_{\infty} $-norms is Max-SNP-hard when $ d $ is part of the input, implying no polynomial-time approximation scheme exists unless P = NP.
• Despite the simplicity of the rectilinear case in 2D, the Euclidean case in the plane remains open, though conjectured to be NP-hard due to structural similarities in the underlying geometry.