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A near-optimal approximation algorithm for Asymmetric TSP on embedded graphs

We present a near-optimal polynomial-time approximation algorithm for the asymmetric traveling salesman problem for graphs of bounded orientable or non-orientable genus. Given any algorithm that achieves an approximation ratio of f(n) on arbitrary n-vertex graphs as a black box, our algorithm achieves an approximation factor of O(f(g)) on graphs with genus g. In particular, the O(log n/loglog n)-approximation algorithm for general graphs by Asadpour et al. [SODA 2010] immediately implies an O(log g/loglog g)-approximation algorithm for genus-g graphs. Moreover, recent results on approximating the genus of graphs imply that our O(log g/loglog g)-approximation algorithm can be applied to bounded-degree graphs even if no genus-g embedding of the graph is given. Our result improves and generalizes the o(√ g log g)-approximation algorithm of Oveis Gharan and Saberi [SODA 2011], which applies only to graphs with orientable genus g and requires a genus-g embedding as part of the input, even for bounded-degree graphs. Finally, our techniques yield a O(1)-approximation algorithm for ATSP on graphs of genus g with running time 2O(g) · nO(1).