[Paper Review] Statistical distributions of quasi-optimal paths in the traveling salesman problem: the role of the initial distribution of cities

H. Hernández-Saldaña, M. Gómez-Quezada+1 more|Redalyc (Universidad Autónoma del Estado de México)|Mar 20, 2010

Mathematical Dynamics and Fractals1 references3 citations

TL;DR

This paper investigates how the initial spatial distribution of cities affects the statistical properties of quasi-optimal traveling salesman problem (TSP) paths. Using perturbed lattice models with uniform and Gaussian noise, it shows that as city distribution randomness increases, the nearest-neighbor spacing of path lengths transitions toward daisy models of rank 2 or 3, with a key transition occurring when perturbations overlap periodic lattice sites—linking these patterns to real-world corporate vote distributions in Mexico.

ABSTRACT

Solutions to Traveling salesman problem have been obtained with several algorithms. However, few of them has discussed about the statistical distribution of lengths for the quasi-optimal path obtained. For a random set of cities such a distribution follows a rank 2 daisy model but our analysis on actual distribution of cities does not show the characteristic quadratic growth of the daisy model. The role played by the initial city distribution is explored in this work. The importance of understanding such a behavior in the context of electoral processes is discussed.

Motivation & Objective

• To understand why real-world TSP solutions do not follow the universal daisy model of rank 2 seen in random city distributions.
• To investigate how initial city distributions—specifically periodic lattices perturbed by random noise—affect the statistical properties of quasi-optimal TSP paths.
• To determine the transition point at which statistical behavior shifts from non-universal to universal-like (daisy model) behavior.
• To explore potential connections between TSP path length statistics and real-world phenomena such as corporate vote distributions in Mexican elections.
• To assess whether global country maps exhibit universal statistical properties or are governed by initial configuration-specific dynamics.

Proposed method

• Model 1: A rectangular grid of cities is perturbed using a uniform random distribution with width σ, simulating city placement variation.
• Model 2: A Gaussian random perturbation is applied to the same grid, with σ as the standard deviation.
• Use of polynomial fitting to separate secular (system-specific) and fluctuation (universal) components of the cumulative path length distribution.
• Apply the Σ² statistics to analyze level spacing fluctuations and compare them to daisy model predictions.
• Map TSP path length distributions to vote distribution data, particularly for Mexican corporate elections (2000–2006), to test for statistical similarity.
• Use of daisy models of rank r to fit the nearest-neighbor spacing of path lengths, with r = 2 and r = 3 as key reference points.

Experimental results

Research questions

• RQ1 Why do real-world TSP maps for countries not exhibit the characteristic quadratic growth of the rank-2 daisy model, unlike randomly distributed cities?
• RQ2 At what level of perturbation (σ) does the statistical behavior of a perturbed lattice transition from non-universal to daisy model-like behavior?
• RQ3 What is the significance of the overlap between perturbation distributions and the original lattice spacing in triggering the emergence of daisy models?
• RQ4 Can the statistical structure of TSP path lengths be meaningfully linked to real-world social phenomena such as corporate vote distributions?
• RQ5 Do global TSP maps for countries exhibit universal statistical properties, or are they dominated by initial city distribution specifics?

Key findings

• For randomly distributed cities, the nearest-neighbor spacing of path lengths follows a daisy model of rank 2, as expected from prior work.
• In contrast, actual country maps do not show the characteristic quadratic growth of the rank-2 daisy model, indicating non-universal behavior.
• For Model I (uniform perturbation), the daisy model of rank 3 emerges when σ = b (where b is the lattice spacing), indicating overlap of perturbation distributions.
• For Model II (Gaussian perturbation), the rank-3 daisy model fits when σ = b/2, corresponding to two standard deviations covering the lattice spacing.
• The Σ² statistics for Model I at σ = 15.0 shows a slope of 0.317486 ± 7.31×10⁻⁵, close to the theoretical 1/3 value for a rank-2 daisy model.
• A strong statistical link is observed between the tail decay of TSP path length distributions and the distribution of corporate votes in Mexican elections (2006), both fitting daisy models of rank 3.