An Olog n/log log n-Approximation Algorithm for the Asymmetric Traveling Salesman Problem

We present a randomized O log n /log log n -approximation algorithm for the asymmetric traveling salesman problem ATSP. This provides the first asymptotic improvement over the long-standing Îï log n -approximation bound stemming from the work of Frieze et al. 1982 [Frieze AM, Galbiati G, Maffioki F 1982 On the worst-case performance of some algorithms for the asymmetric traveling salesman problem. Networks 121:23-39]. The key ingredient of our approach is a new connection between the approximability of the ATSP and the notion of so-called thin trees. To exploit this connection, we employ maximum entropy rounding-a novel method of randomized rounding of LP relaxations of optimization problems. We believe that this method might be of independent interest.