The asymmetric traveling salesman path LP has constant integrality ratio

Abstract

We show that the classical LP relaxation of the asymmetric traveling salesman path problem (ATSPP) has constant integrality ratio. If ρATSP and ρATSPP denote the integrality ratios for the asymmetric TSP and its path version, then ρATSPP≤ 4 ρATSP- 3. We prove an even better bound for node-weighted instances: if the integrality ratio for ATSP on node-weighted instances is ρATSPNW, then the integrality ratio for ATSPP on node-weighted instances is at most 2ρATSPNW-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.