Hard to solve instances of the Euclidean Traveling Salesman Problem

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 more runtime than 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.

[1]  Ronald L. Graham,et al.  Some NP-complete geometric problems , 1976, STOC '76.

[2]  George B. Dantzig,et al.  Solution of a Large-Scale Traveling-Salesman Problem , 1954, Oper. Res..

[3]  Günter Rote,et al.  The Convex-Hull-and-Line Traveling Salesman Problem: A Solvable Case , 1994, Inf. Process. Lett..

[4]  L. Wolsey.  Heuristic analysis, linear programming and branch and bound , 1980 .

[5]  Richard M. Karp,et al.  Reducibility Among Combinatorial Problems , 1972, 50 Years of Integer Programming.

[6]  Nicos Christofides.  Worst-Case Analysis of a New Heuristic for the Travelling Salesman Problem , 1976, Operations Research Forum.

[7]  M. M. Flood.  The Traveling-Salesman Problem , 1956 .

[8]  Stefan Hougardy.  On the integrality ratio of the subtour LP for Euclidean TSP , 2014, Oper. Res. Lett..

[9]  Sanjeev Arora,et al.  Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems , 1998, JACM.

[10]  William J. Cook,et al.  The Traveling Salesman Problem: A Computational Study , 2007 .

[11]  Christos H. Papadimitriou,et al.  The Euclidean Traveling Salesman Problem is NP-Complete , 1977, Theor. Comput. Sci..

[12]  William J. Cook,et al.  Solution of a Large-Scale Traveling-Salesman Problem , 1954, 50 Years of Integer Programming.

[13]  Nicholas C. Wormald,et al.  A polynomial algorithm for a constrained traveling salesman problem , 2001, Networks.

[14]  David P. Williamson.  ANALYSIS OF THE HELD-KARP HEURISTIC FOR THE TRAVELING SALESMAN PROBLEM , 1990 .

[15]  Martin Grötschel,et al.  The ellipsoid method and its consequences in combinatorial optimization , 1981, Comb..