[Paper Review] The Approximation Ratio of the $k$-Opt Heuristic for the Euclidean Traveling Salesman Problem

Ulrich A. Brodowsky, Stefan Hougardy+1 more|arXiv (Cornell University)|Aug 31, 2021

Vehicle Routing Optimization Methods15 references4 citations

TL;DR

This paper establishes the approximation ratio of the $k$-Opt heuristic for the Euclidean Traveling Salesman Problem as $\Theta(\log n / \log\log n)$ for constant $k \geq 2$, resolving a long-standing open problem by providing the first non-trivial lower bound for $k \geq 3$ and improving the upper bound for $k=2$. The result holds for all $p$-norms with $1 \leq p < \infty$, using planarity arguments and geometric analysis of 2-optimal tours.

ABSTRACT

The $k$-Opt heuristic is a simple improvement heuristic for the Traveling Salesman Problem. It starts with an arbitrary tour and then repeatedly replaces $k$ edges of the tour by $k$ other edges, as long as this yields a shorter tour. We will prove that for 2-dimensional Euclidean Traveling Salesman Problems with $n$ cities the approximation ratio of the $k$-Opt heuristic is $Θ(\log n / \log \log n)$. This improves the upper bound of $O(\log n)$ given by Chandra, Karloff, and Tovey in 1999 and provides for the first time a non-trivial lower bound for the case $k\ge 3$. Our results not only hold for the Euclidean norm but extend to arbitrary $p$-norms with $1 \le p &lt; \infty$.

Motivation & Objective

• To close the gap in understanding the approximation ratio of the $k$-Opt heuristic for the Euclidean TSP, particularly for $k \geq 3$, where no non-trivial lower bound was previously known.
• To improve the upper bound for the 2-Opt heuristic from $O(\log n)$ to $O(\log n / \log\log n)$, matching the known lower bound.
• To extend the approximation ratio results to all $p$-norms with $1 \leq p < \infty$, not just the Euclidean norm.
• To prove that the $k$-Opt heuristic for $k \geq 2$ always returns a 2-optimal solution, enabling the use of 2-Opt bounds for $k$-Opt analysis.

Proposed method

• Reduction of the 2-Opt approximation ratio to the case of 2-optimal tours with no edge crossings, leveraging planarity and geometric properties.
• Construction of a family of Euclidean TSP instances with $n$ points where the optimal tour is significantly shorter than any 2-optimal tour, using a 3D grid-like structure embedded in the plane.
• Use of planar graph theory and edge-disjoint path decomposition to bound the number of intersecting edges in 2-optimal tours.
• Application of geometric arguments based on axis-aligned rectangles and edge orientations to handle the $L^1$-norm case, where crossings can occur in optimal tours.
• Proof that in $L^1$-norm, 2-optimal tours cannot contain non-axis-aligned crossing edges, allowing decomposition into three non-crossing edge sets.
• Extension of the lower bound construction to arbitrary $p$-norms for $p > 1$ via preservation of planarity and geometric inequalities.

Experimental results

Research questions

• RQ1 What is the exact approximation ratio of the $k$-Opt heuristic for the Euclidean TSP when $k \geq 3$, given that no non-trivial lower bound was previously known?
• RQ2 Can the upper bound of $O(\log n)$ for the 2-Opt heuristic be improved, and if so, to what tight asymptotic value?
• RQ3 Does the approximation ratio of the $k$-Opt heuristic remain $\Theta(\log n / \log\log n)$ when using $p$-norms instead of the Euclidean norm?
• RQ4 How do crossings in 2-optimal tours affect the approximation ratio in the $L^1$-norm, and can they be handled without losing the asymptotic bound?

Key findings

• The approximation ratio of the $k$-Opt heuristic for the Euclidean TSP with $n$ cities is $\Theta(\log n / \log\log n)$ for all constant $k \geq 2$, resolving a long-standing open problem.
• The upper bound for the 2-Opt heuristic is improved from $O(\log n)$ to $O(\log n / \log\log n)$, matching the previously known lower bound.
• A non-trivial lower bound of $\Omega(\log n / \log\log n)$ is established for the $k$-Opt heuristic when $k \geq 3$, which was previously unknown.
• The result extends to all $p$-norms with $1 \leq p < \infty$, including the $L^1$-norm, where a special argument is used to handle crossings in 2-optimal tours.
• For the $L^1$-norm, it is shown that 2-optimal tours cannot contain non-axis-aligned crossing edges, allowing decomposition into three non-crossing edge sets with bounded total length.
• The final approximation ratio for $k$-Opt under any $p$-norm with $1 \leq p < \infty$ is $\Theta(\log n / \log\log n)$, with a loss factor of at most 3 in the $L^1$ case.