An O(logn) Approximation Ratio for the Asymmetric Traveling Salesman Path Problem

Abstract

Given an arc-weighted directed graph G = (V,A,ℓ) and a pair of vertices s,t, we seek to find an s-twalk of minimum length that visits all the vertices in V. If ℓ satisfies the asymmetric triangle inequality, the problem is equivalent to that of finding an s-tpath of minimum length that visits all the vertices. We refer to this problem as ATSPP. When s = t this is the well known asymmetric traveling salesman tour problem (ATSP). Although an O(logn) approximation ratio has long been known for ATSP, the best known ratio for ATSPP is $O(\sqrt{n})$. In this paper we present a polynomial time algorithm for ATSPP that has approximation ratio of O(logn). The algorithm generalizes to the problem of finding a minimum length path or cycle that is required to visit a subset of vertices in a given order.