On the Approximation Ratio of the $k$-Opt and Lin-Kernighan Algorithm

TL;DR

This work investigates the worst-case performance of local-search-based TSP heuristics, showing that for fixed $k\ge3$ the $k$-Opt and (via a parameterized Lin–Kernighan variant) yield an approximation ratio of $O(n^{1/k})$ on Metric TSP. By tying edge-length structure to extremal graph theory quantities $\mathrm{ex}(n,2k)$, it derives conditional lower bounds (under the Erdős girth conjecture) and unconditional results for key $k$ values, with a tight characterization for $k=3,4,6$ up to logarithmic factors. It extends these upper-bound techniques to Graph TSP and establishes a near $\Theta(\log n/\log\log n)$ lower bound, plus a matching (up-to-logs) upper bound for the 2-Opt family. For the (1,2)-TSP, the paper proves a universal lower bound of $11/10$ on the approximation ratio of $k$-improv and $k$-Opt across fixed $k$, illustrating fundamental limits of these local-search strategies. Collectively, the results sharpen our understanding of how local improvement depth and graph girth constrain worst-case performance in both Metric and unweighted-graph TSP settings.

Abstract

The $k$-Opt and Lin-Kernighan algorithm are two of the most important local search approaches for the Metric TSP. Both start with an arbitrary tour and make local improvements in each step to get a shorter tour. We show that for any fixed $k\geq 3$ the approximation ratio of the $k$-Opt algorithm for Metric TSP is $O(\sqrt[k]{n})$. Assuming the Erdős girth conjecture, we prove a matching lower bound of $Ω(\sqrt[k]{n})$. Unconditionally, we obtain matching bounds for $k=3,4,6$ and a lower bound of $Ω(n^{\frac{2}{3k-3}})$. Our most general bounds depend on the values of a function from extremal graph theory and are tight up to a factor logarithmic in the number of vertices unconditionally. Moreover, all the upper bounds also apply to a parameterized generalization of the Lin-Kernighan algorithm with appropriate parameters. We also show that the approximation ratio of $k$-Opt for Graph TSP is $Ω\left(\frac{\log(n)}{\log\log(n)}\right)$ and $O\left(\left(\frac{\log(n)}{\log\log(n)}\right)^{\log_2(9)+ε}\right)$ for all $ε>0$. For the (1,2)-TSP we give a lower bound of $\frac{11}{10}$ on the approximation ratio of the $k$-improv and $k$-Opt algorithm for arbitrary fixed $k$.

Figures (19)

• Figure 1: Assume that a tour $T$ contains the edges $(a,b)$ and $(c,d)$ whose corresponding shortest paths are depicted here as the straight edges for $(a,b)$ and dashed edges for $(c,d)$. If the shortest paths share a common edge, then the 2-move replacing the two edges by $\{a,c\}$ and $\{b,d\}$ would be improving.
• Figure 2: Apply an improving $k$-move to the tour $T'$ of the instance $I'$. After contracting the vertices $v$ and $v'$, the result $T_2$ may visit $v$ multiple times.
• Figure 3: Shortcut $\{a_1,v\}$ and $\{v,a_2\}$ to $\{a_1,a_2\}$ in $T_2$ to get a tour. To directly construct this tour from $T$ make the following modifications to the $k$-move: Replace all occurrences of $v'$ by $v$. Instead of deleting $\{v,v\}$ we delete $T\cap\{\{a_1,v\}, \{v,a_2\}\}$. Instead of adding $\{\{a_1,v\}, \{v,a_2\}\} \backslash T$ we add $\{a_1,a_2\}$.
• Figure 4: An example instance with a $k$-optimal tour, i.e. the directed graph $G$. The blue and red edges are the $C$-edges and connecting path edges that arise from the chosen cycle in $G_2$ in Figure \ref{['g2']}, respectively. Note that the optimal tour is not drawn here, so it is not clear from the figure which vertices to contract to construct $G_1$.
• Figure 5: The directed multigraph $G_1$: We contracted vertices that lie near each other in the optimal tour. We can see there are 4 pairs of vertices in Figure \ref{['tour']} that were contracted.

Theorems & Definitions (124)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Corollary 1.4
• Corollary 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9
• Theorem 1.10