Worst Case and Probabilistic Analysis of the 2-Opt Algorithm for the TSP (2302.06889v1)

Abstract: 2-Opt is probably the most basic local search heuristic for the TSP. This heuristic achieves amazingly good results on real world Euclidean instances both with respect to running time and approximation ratio. There are numerous experimental studies on the performance of 2-Opt. However, the theoretical knowledge about this heuristic is still very limited. Not even its worst case running time on 2-dimensional Euclidean instances was known so far. We clarify this issue by presenting, for every $p\in\mathbb{N}$, a family of $L_p$ instances on which 2-Opt can take an exponential number of steps. Previous probabilistic analyses were restricted to instances in which $n$ points are placed uniformly at random in the unit square $[0,1]^2$. We consider a more advanced model in which the points can be placed independently according to general distributions on $[0,1]^d$, for an arbitrary $d\ge 2$. In particular, we allow different distributions for different points. We study the expected number of local improvements in terms of the number $n$ of points and the maximal density $\phi$ of the probability distributions. We show an upper bound on the expected length of any 2-Opt improvement path of $\tilde{O}(n^{4+1/3}\cdot\phi^{8/3})$. When starting with an initial tour computed by an insertion heuristic, the upper bound on the expected number of steps improves even to $\tilde{O}(n^{4+1/3-1/d}\cdot\phi^{8/3})$. If the distances are measured according to the Manhattan metric, then the expected number of steps is bounded by $\tilde{O}(n^{4-1/d}\cdot\phi)$. In addition, we prove an upper bound of $O(\sqrt[d]{\phi})$ on the expected approximation factor with respect to all $L_p$ metrics. Let us remark that our probabilistic analysis covers as special cases the uniform input model with $\phi=1$ and a smoothed analysis with Gaussian perturbations of standard deviation $\sigma$ with $\phi\sim1/\sigma^d$.

• The paper presents a worst-case construction showing that 2-Opt can require an exponential number of steps in specific TSP instances.
• The paper introduces a probabilistic model that derives bounds on the expected running time and improvement steps over diverse d-dimensional vertex distributions.
• The paper demonstrates practical implications by establishing performance limits on ϕ-perturbed Manhattan and Euclidean instances, informing TSP applications.

Analyzing the Worst Case and Probabilistic Performance of the 2-Opt Heuristic for TSP

Overview

The paper under review explores the theoretical underpinnings of the 2-Opt algorithm, a well-renowned heuristic for the Traveling Salesman Problem (TSP). Despite the empirical success of 2-Opt in generating satisfactory solutions for Euclidean instances, its worst-case performance and theoretical behavior are not as well-understood. Hence, the authors focus on elucidating these aspects through a mix of worst-case scenarios and probabilistic analyses, presenting bounds on the expected number of steps 2-Opt might take, its running time, and approximation ratio.

Main Contributions

• Worst-Case Construction: The authors present instances in which 2-Opt takes an exponential number of steps. This analysis spans various metrics, including the $L_p$ metrics, addressing an open question in the domain regarding whether an exponential number of steps could be required in Euclidean instances.
• Probabilistic Model: Building on previous probabilistic analyses, the paper explores instances where vertices are chosen independently according to different distributions, broadening the horizon from previous analyses which only considered uniformly random distributions within a unit square. Additionally, the authors propose bounds for general distributions in $d$-dimensional spaces, providing insights into how parameters like maximal density $\phi$ affect the algorithm's performance.
• Divergent Implications: The presented theoretical foundations offer an evaluative perspective on the operational efficiency and limits of 2-Opt when faced against both artificially constructed worst-case instances and more realistic, probabilistic occurrences in application scenarios.

Numerical Results and Implications

The paper puts forth strong numerical results showcasing the bounds on improvement paths. For instance, on $\phi$-perturbed Manhattan instances, the expected length of the path in the 2-Opt state graph is bounded by $\tilde{O}(n^{4}\cdot\phi)$. Similarly, a more complex upper bound of $$\tilde{O}(n^{4+1/3}\cdot\log(n\phi)\cdot\phi^{8/3})$$ is derived for $\phi$-perturbed Euclidean instances.

In addition to these bounds, the authors explore the expected approximation ratios of results generated by 2-Opt on $\phi$-perturbed $L_p$ instances. They reveal that the expected ratio is $O(\sqrt[d]{\phi})$, potentially illuminating practical relevance for scenarios involving Euclidean spaces.

Theoretical and Practical Implications

On the theoretical front, the paper expands the understanding of local search in high-dimensional Euclidean spaces, identifying mathematical dynamics across probabilistic models and worst-case configurations. Practically, insights gathered could prove invaluable in applications ranging from logistics, chip design, and any computational setting relying on TSP algorithms, especially those adopting local search as a solution strategy.

Future Directions

Building on its achievements, the paper signals several avenues for future inquiry. These include refining the probabilistic analysis to close the gap between theory and empirical observations further and extending similar analytic frameworks to other local search strategies or heuristic paradigms.

This exhaustive investigation into 2-Opt under diverse theoretical lenses provides a robust backdrop against which future tweaks and innovations in heuristic design can be gauged, essentially enriching the narrative on the Traveling Salesman Problem and local searches broadly.