An improved integrality gap for asymmetric TSP paths

The Asymmetric Traveling Salesperson Path (ATSPP) problem is one where, given an asymmetric metric space (V,d) with specified vertices s and t, the goal is to find an s-t path of minimum length that visits all the vertices in V. This problem is closely related to the Asymmetric TSP (ATSP) problem, which seeks to find a tour (instead of an s-t path) visiting all the nodes: for ATSP, a ρ-approximation guarantee implies an O(ρ)-approximation for ATSPP. However, no such connection is known for the integrality gaps of the linear programming relxations for these problems: the current-best approximation algorithm for ATSPP is O(logn/loglogn), whereas the best bound on the integrality gap of the natural LP relaxation (the subtour elmination LP) for ATSPP is O(logn). In this paper, we close this gap, and improve the current best bound on the integrality gap from O(logn) to O(logn/loglogn). The resulting algorithm uses the structure of narrow s-t cuts in the LP solution to construct a (random) tree witnessing this integrality gap. We also give a simpler family of instances showing the integrality gap of this LP is at least 2.