Short combinatorial proof that the DFJ polytope is contained in the MTZ polytope for the Asymmetric Traveling Salesman Problem

Abstract For the Asymmetric Traveling Salesman Problem (ATSP), it is known that the Dantzig–Fulkerson–Johnson (DFJ) polytope is contained in the Miller–Tucker–Zemlin (MTZ) polytope. The analytic proofs of this fact are quite long. Here, we present a proof which is combinatorial and significantly shorter by relating the formulation to distances in a modified graph.