Report on article The Travelling Salesman Problem: A Linear Programming Formulation

Unknown relation between P and NP [3] complexity classes remains to be one of significant non solved problems in complexity theory. P complexity class consists of problems solvable by Deterministic Turing Machine (DTM) in polynomially bounded time, while NP complexity class consists of problem solvable by Non Deterministic Turing Machine (NDTM) in polynomially bounded time. This means that DTM can verify solution of every NP problem in polynomially bounded time even if polynomial algorithm for finding this solution is unknown [16]. Significant subclass of NP problems is known as NP-complete class. Problems from this class have ability to represent any other problems from whole NP complexity class. In 1970 S. Cook presented in [2] first reduction from any NP problem to Boolean Satisfiability Problem, and two years after R. Karp proved that 21 other problems are in NP-complete class showing many-one polynomial time reductions to these problems [14]. If then anyone shows algorithm solving any NP-complete problem in polynomially bounded time then any of NP problems may be solved in no more then O(n) steps, where n stands for instance size and c is some constant value [16]. In 2006 there appeared articles claiming to have proven that P=NP formulating TSP and QAP problem in terms of linear programming ( [4], [10], [5] and other). Author of this article have prepared counter example [11] for one of these article discussing inability for LP approach to solve large instances of NP-complete problems [12]. Some suggestions from [11] were taken into proposed model and counter example is not valid for new version of these models. In this article we present extended version proving flaws in extended model.