Scalefree hardness of average-case Euclidean TSP approximation

We show that if P$\neq$NP, then a wide class of TSP heuristics fail to approximate the length of the TSP to asymptotic optimality, even for random Euclidean instances. Previously, this result was not even known for any heuristics (greedy, etc) used in practice. As an application, we show that when using a heuristic from this class, a natural class of branch-and-bound algorithms takes exponential time to find an optimal tour (again, even on a random point-set), regardless of the particular branching strategy or lower-bound algorithm used.

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

[2]  J. Beardwood,et al.  The shortest path through many points , 1959, Mathematical Proceedings of the Cambridge Philosophical Society.

[3]  David S. Johnson,et al.  Asymptotic experimental analysis for the Held-Karp traveling salesman bound , 1996, SODA '96.

[4]  Alan M. Frieze,et al.  Separating subadditive euclidean functionals , 2017, Random Struct. Algorithms.

[5]  S. Steinerberger.  New Bounds for the Traveling Salesman Constant , 2013, Advances in Applied Probability.

[6]  J. Steele.  Probability theory and combinatorial optimization , 1987 .

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

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

[9]  Eugene L. Lawler,et al.  Traveling Salesman Problem , 2016 .

[10]  George L. Nemhauser,et al.  The Traveling Salesman Problem: A Survey , 1968, Oper. Res..

[11]  Juliane Jung,et al.  The Traveling Salesman Problem: A Computational Study , 2007 .

[12]  J. Steele.  Subadditive Euclidean Functionals and Nonlinear Growth in Geometric Probability , 1981 .

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

[14]  Richard M. Karp,et al.  The traveling-salesman problem and minimum spanning trees: Part II , 1971, Math. Program..

[15]  L. Fejes.  Über einen geometrischen Satz , 1940 .

[16]  Richard M. Karp,et al.  The Traveling-Salesman Problem and Minimum Spanning Trees , 1970, Oper. Res..

[17]  M. Held,et al.  A dynamic programming approach to sequencing problems , 1962, ACM National Meeting.

[18]  R. Jonker,et al.  A branch and bound algorithm for the symmetric traveling salesman problem based on the 1-tree relaxation , 1982 .

[19]  Joseph S. B. Mitchell,et al.  Guillotine Subdivisions Approximate Polygonal Subdivisions: A Simple Polynomial-Time Approximation Scheme for Geometric TSP, k-MST, and Related Problems , 1999, SIAM J. Comput..

[20]  Colin McDiarmid,et al.  Surveys in Combinatorics, 1989: On the method of bounded differences , 1989 .

[21]  L. Few.  The shortest path and the shortest road through n points , 1955 .

[22]  Donald L. Miller,et al.  Exact Solution of Large Asymmetric Traveling Salesman Problems , 1991, Science.

[23]  G. A. Miller,et al.  MATHEMATISCHE ZEITSCHRIFT. , 1920, Science.