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

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. Scutella [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.