On Lovász--Schrijver Lift-and-Project Procedures on the Dantzig--Fulkerson--Johnson Relaxation of the TSP

We study the Lovasz--Schrijver lift-and-project procedure $N_+$ on the linear relaxation of the Dantzig--Fulkerson--Johnson formulation of the traveling salesman problem (TSP). A long standing conjecture states that the integrality gap of this relaxation is $\frac43$ in the case of metric costs. In this paper, we show that the $N_+$-rank of 2-matching inequalities relative to this relaxation can be arbitrarily high and obtain as a corollary that even after applying $N_+$ to the relaxation a fixed number of times, the integrality gap of the resulting relaxation is at least $\frac43$.