Symmetric traveling salesman problem and flows in hypergraphs: New algorithmic possibilities

Authors

  • Tiru S Arthanari The University of Auckland

DOI:

https://doi.org/10.1478/AAPP.971A1

Keywords:

pedigree polytope, multistage insertion formulation, symmetric traveling salesman problem, minimum cost hypergraph flow problem, Leontief substitution flow problem, Lagrangean relaxation, Lagrangean decomposition

Abstract

The traveling salesperson problem (TSP) is a very well-known NP-hard combinatorial optimization problem. Many different integer programming formulations are known for the TSP. Some of them are “compact”, in the sense of having a polynomially-bounded number of variables and constraints. We focus on one compact formulation for the symmetric TSP, known as the “multistage insertion” formulation [T. S. Arthanari, Discrete Mathematics 306, 1474 (2006)], and show that its linear programming (LP) relaxation can be viewed as a minimum-cost flow problem in a hypergraph. Using some ideas of R. Cambini, G. Gallo and M. G. Scutellà [University of Pisa, TR-1/92 (1992)] on flows in hypergraphs, we propose new algorithms to solve the LP relaxation. We also exploit the Leontief structure of a certain subproblem, to provide additional algorithms for solving the LP relaxation. Some bounds on the running time are also derived.

Author Biography

  • Tiru S Arthanari, The University of Auckland
    Department of Information Systems and Operations Management

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Published

2019-01-10

Issue

Vol. 97 No. 1 (2019)

Section

Articles

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