The asymmetric traveling salesman path LP has constant integrality ratio

We show that the classical LP relaxation of the asymmetric traveling salesman path problem (ATSPP) has constant integrality ratio. If $\rho_{\text{ATSP}}$ and $\rho_{\text{ATSPP}}$ denote the integrality ratios for the asymmetric TSP and its path version, then $\rho_{\text{ATSPP}}\le 4\rho_{\text{ATSP}}-3$. We prove an even better bound for node-weighted instances: if the integrality ratio for ATSP on node-weighted instances is $\rho_{\text{ATSP}}^{\text{NW}}$, then the integrality ratio for ATSPP on node-weighted instances is at most $2\rho_{\text{ATSP}}^{\text NW}-1$. Moreover, we show that for ATSP node-weighted instances and unweighted digraph instances are almost equivalent. From this we deduce a lower bound of 2 on the integrality ratio of unweighted digraph instances.