Abtract. 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 \(O(n^9) \times O(n^8)\) model*. Here we have only \(O(n^5)\) variables and \(O(n^4)\) 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 solvable in polynomial time 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 extended formulations results (such as Braun et al. (2015) or Fiorini et al. (2015) in particular) are briefly discussed in an appendix. [*: Advances in Combinatorial Optimization: Linear Programming Formulation of the Traveling Salesman and Other Hard Combinatorial Optimization Problems (World Scientific, January 2016).]
https://arxiv.org/abs/1610.00353
Unfortunately there are about as many \(P=NP\) proofs around as there are \(P\ne NP\) proofs. See (1) for a large list.
References
- Gerhard J. Woeginger, The P-versus-NP page, http://www.win.tue.nl/~gwoegi/P-versus-NP.htm