Stefan Hougardy

Sophia Heimann, Hung P. Hoang, Stefan Hougardy

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.

William Cook, Stefan Hougardy, Moritz Petrich

The subtour relaxation of the traveling salesman problem (TSP) plays a central role in approximation algorithms and polyhedral studies of the TSP. A long-standing conjecture asserts that the integrality gap of the subtour relaxation for the metric TSP is exactly 4/3. In this paper, we extend the exact verification of this conjecture for small numbers of vertices. Using the framework introduced by Benoit and Boyd in 2008, we confirm their results up to n=10. We further show that for n=11 and n=12, the published lists of extreme points of the subtour polytope are incomplete: one extreme point is missing for n=11 and twenty-two extreme points are missing for n=12. We extend the enumeration of the extreme points of the subtour polytope to instances with up to 14 vertices in the general case. Restricted to half-integral vertices, we extend the enumeration of extreme points up to n=17. Our results provide additional support for the 4/3-Conjecture.