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