The $k$-Opt algorithm for the Traveling Salesman Problem has exponential running time for $k \ge 5$

TL;DR

The paper tackles the hardness of the $k$-Opt local search for the Traveling Salesman Problem, proving that the algorithm exhibits exponential running time for all pivot rules when $k \ge 5$, and that TSP/$k$-Opt is PLS-complete for $k \ge 17$. The authors construct a sequence of reductions from Max-Cut/Flip using parity and XOR gadgets, along with a careful gadget equipping and labeling scheme, to translate flipping in Max-Cut into $k$-swaps in TSP. They progressively tighten the thresholds—from $k \ge 13$ to $k \ge 9$ and finally to $k \ge 5$—by leveraging the Michel–Scott construction, introducing a flexible gadget and a double gadget, and by controlling local strictness properties. The results settle open questions about the practical limits of local-search hardness in TSP and offer a rigorous PLS-completeness proof for relatively small $k$, with implications for understanding the limits of local-search heuristics in combinatorial optimization.

Abstract

The $k$-Opt algorithm is a local search algorithm for the Traveling Salesman Problem. Starting with an initial tour, it iteratively replaces at most $k$ edges in the tour with the same number of edges to obtain a better tour. Krentel (FOCS 1989) showed that the Traveling Salesman Problem with the $k$-Opt neighborhood is complete for the class PLS (polynomial time local search) and that the $k$-Opt algorithm can have exponential running time for any pivot rule. However, his proof requires $k \gg 1000$ and has a substantial gap. We show the two properties above for a much smaller value of $k$, addressing an open question by Monien, Dumrauf, and Tscheuschner (ICALP 2010). In particular, we prove the PLS-completeness for $k \geq 17$ and the exponential running time for $k \geq 5$.

Figures (13)

• Figure 1: The first-set edge $x_{\ell} x_r$ and the second-set path $(x_{\ell}, x_1, x'_1, x_2, x'_2, x_3, x'_3, x_r)$ of an $H$-vertex $x$ of degree three. The dashed edges are gateways. The other edges of the second-set path are doors.
• Figure 2: An example of our reduction from a Max-Cut instance (left figure) to a TSP instance (right figure). The parity gadgets are indicated by the blue circles attached to three edges each. The XOR gadgets are indicated by red boxes attached to two edges each.
• Figure 3: An example of a parity gadget (left figure). The right four figures show the four possibilities subtour (1)--(4) to cover the vertices of the parity gadget by disjoint paths (red edges and red endpoints). The dashed edges are the same-set edges, the dotted edges are the different-set edges, and the solid edges are the remaining edges.
• Figure 4: The XOR gadget of order four (a) and its two subtours ((b) and (c)). (d)-(f) are examples of assigning the XOR gadgets of order from zero to two to the $H$-vertex $x$, where dashed edges, dotted edges, and bold edges represent the left first-set edge, the door closest to $x_r$, and the right first-set edge, respectively.
• Figure 5: (a) A (4,2)-simple gadget. Bold red edges are incident to degree-two vertices. Dashed edges are external edges. $Y'a$ and $ZZ'$ are same-set edges with same-set weight $\sigma$, while $Z'a$ and $Y'Z$ are different-set edges with different-set weight $\delta$. (b)-(e) show the subtours (1)-(4).

Theorems & Definitions (44)

• Theorem 1
• Theorem 2
• Definition 3: All-exp property
• Definition 4: The class PLS JPY1988
• Definition 5: PLS-completeness JPY1988
• Theorem 6: Monien_Max_Cut_4Michel_Max_Cut_4
• Definition 7: Parity Gadget
• Definition 8: XOR Gadget
• Lemma 10
• proof