On modeling NP-Complete problems as polynomial-sized linear programs: Escaping/Side-stepping the "barriers"

In view of the extended formulations (EFs) developments (e.g. "Fiorini, S., S. Massar, S. Pokutta, H.R. Tiwary, and R. de Wolf [2015]. Exponential Lower Bounds for Polytopes in Combinatorial Optimization. Journal of the ACM 62:2"), we focus in this paper on the question of whether it is possible to model an NP-Complete problem as a polynomial-sized linear program. For the sake of simplicity of exposition, the discussions are focused on the TSP. We show that a finding that there exists no polynomial-sized extended formulation of "the TSP polytope" does not (necessarily) imply that it is "impossible" for a polynomial-sized linear program to solve the TSP optimization problem. We show that under appropriate conditions the TSP optimization problem can be solved without recourse to the traditional city-to-city ("travel leg") variables, thereby side-stepping/"escaping from" "the TSP polytope" and hence, the barriers. Some illustrative examples are discussed.