A Small-Order-Polynomial-Sized Linear Program for Solving the Traveling Salesman Problem

We present a polynomial-sized linear program (LP) for the n-city TSP drawing upon “complex flow” modeling ideas by the first two authors who used an O(n)×O(n) model*. Here we have only O(n) variables and O(n) constraints. We do not model explicit cycles of the cities, and our modeling does not involve the city-to-city variables-based, traditional TSP polytope referred to in the literature as “The TSP Polytope.” Optimal TSP objective value and tours are achieved by solving our proposed LP. In the case of a unique optimum, the integral solution representing the optimal tour is obtained using any LP solver (solution algorithm). In the case of alternate optima, an LP solver (e.g., an interior-point solver) may stop with a fractional (interior-point) solution, which (we prove) is a convex combination of alternate optimal TSP tours. In such cases, one of the optimal tours can be trivially retrieved from the solution using a simple iterative elimination procedure we propose. We have solved over a million problems with up to 27 cities using the barrier methods of CPLEX, consistently obtaining all integer solutions. Since LP is polynomially solvable and we have a model which is of polynomial size in n, the paper is thus offering (although, incidentally) a proof of the equality of the computational complexity classes “P ” and “NP ”. The non-applicability and numerical refutations of existing negative extended formulations results (such as Braun et. al. (2015)∗∗ and Fiorini et al. (2015)∗∗∗ in particular) are briefly discussed in an appendix.