A Polynomial Time Algorithm for The Traveling Salesman Problem

The ATSP polytope can be expressed by asymmetric polynomial size linear program. Introduction In his seminal work [1], M. Yannakakis proved that the Traveling Salesman problem (TSP) polytope cannot be expressed by symmetric polynomial size linear program, where symmetry means that the polytope is an invariant under vertex relabeling. The question about the size of asymmetric linear programs was left open in [1], and it remained open since. This article answers the question. We reduce the Asymmetric Traveling Salesman problem (ATSP) to an asymmetric polynomial size linear program whose feasible set is asymmetric in the sense of [1]. 1. Directed Hamiltonian Cycle Problem Directed Hamiltonian Cycle Problem (DHC) is a problem to find in any given digraph a cycle which visits all vertices (a circuit which visits all vertices and visits each of them just once). Any such cycle is called Hamiltonian. Ultimately, the problem can be solved by testing all vertex permutations on being a Hamiltonian cycle, or by “growing” of paths. The only drawback here is the number of “elementary operations” required in worst case. Computational complexity of the methods will be factorial and exponential appropriately. So, they are infeasible for modern computers even in the cases of modest digraphs. The real problem is to detect Hamiltonian cycles in feasible time. Sometimes, that can be done using the appropriate criterion [2, 3, 4, 5, 6, 7, 8, and many others]. But, such particular instances of DHC are rare. The importance of DHC follows from the fact that its decision version is a NPcomplete problem [9]. There is a wide spread belief in P 6= NP. This hypothesis 2000 Mathematics Subject Classification. Primary 68Q15, 68R10, 90C57.