On the PLS-Completeness of $k$-Opt Local Search for the Traveling Salesman Problem

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). However, his proof requires $k \gg 1000$ and has a substantial gap. We provide the first rigorous proof for the PLS-completeness and at the same time drastically lower the value of $k$ to $k \geq 15$, addressing an open question by Monien, Dumrauf, and Tscheuschner (ICALP 2010). Our result holds for both the general and the metric traveling salesman problem.

Figures (9)

• 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: The strict gadget (top left) and the subtours (1)--(4). Bold red edges in the top left are edges always in a subtour; those edges in the other panels show the subtours. Dashed edges are external edges (i.e. edges that are incident to the gadget in the overall construction, but not part of the gadget itself). The 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 $\delta$.
• Figure 4: The flexible gadget (left hand figure). The red edges in the left figure show the four edges that have to be contained in every subtour. The next four figures show the four possibilities for subtour (1) -- subtour (4) to cover the vertices of the flexible gadget by disjoint paths (red edges and red endpoints). The last three figures are the three non-standard subtours. 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 5: The XOR gadget of order four (a) and its two subtours (b) and (c).

Theorems & Definitions (39)

• Theorem 1
• Definition 2: The class PLS JPY1988
• Definition 3: PLS-reduction JPY1988
• Definition 4: Tight PLS-reduction schaffer1991
• Lemma 5
• proof
• Definition 6
• Definition 8: XOR Gadget
• Lemma 10
• proof