An improved integrality gap for asymmetric TSP paths

The asymmetric traveling salesperson path problem (ATSPP) is one where, given an asymmetric metric space (V,5) with specified vertices s and t, the goal is to find an s-t path of minimum length that passes through all the vertices in V. This problem is closely related to the asymmetric TSP (ATSP), 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 (LP) relaxations for these problems: the currentbest approximation algorithm for ATSPP is O(ln n/ln ln n), whereas the best bound on the integrality gap of the natural LP relaxation (the subtour elimination LP) for ATSPP is O(ln n). In this paper, we close this gap, and improve the current best bound on the integrality gap from O(ln n) to O(ln n/ln ln n). The resulting algorithm uses the structure of narrow s-t cuts in the LP solution to construct a (random) tree spanning tree that can be cheaply augmented to contain an Eulerian s-t walk. We also build on a result of Oveis Gharan and Saberi and show a strong form of Goddyn's conjecture about thin spanning trees implies the integrality gap of the subtour elimination LP relaxation for ATSPP is bounded by a constant. Finally, we give a simpler family of instances showing the integrality gap of this LP is at least 2.