Hard to Solve Instances of the Euclidean Traveling Salesman Problem
Abstract
The well known $4/3$ conjecture states that the integrality ratio of the subtour LP is at most $4/3$ for metric Traveling Salesman instances. We present a family of Euclidean Traveling Salesman instances for which we prove that the integrality ratio of the subtour LP converges to $4/3$. These instances (using the rounded Euclidean norm) turn out to be hard to solve exactly with Concorde, the fastest existing exact TSP solver. For a 200 vertex instance from our family of Euclidean Traveling Salesman instances Concorde needs several days of CPU time. This is more than 1,000,000 times the runtime for a TSPLIB instance of similar size. Thus our new family of Euclidean Traveling Salesman instances may serve as new benchmark instances for TSP algorithms.
 Publication:

arXiv eprints
 Pub Date:
 August 2018
 arXiv:
 arXiv:1808.02859
 Bibcode:
 2018arXiv180802859H
 Keywords:

 Computer Science  Discrete Mathematics;
 Computer Science  Computational Complexity;
 Computer Science  Data Structures and Algorithms;
 Mathematics  Combinatorics;
 90C05;
 90C06;
 90C10;
 90C27
 EPrint:
 doi:10.1007/s12532020001845