In the Asymmetric Traveling Salesperson Problem (ATSP) the goal is to find a
closed walk of minimum cost in a directed graph visiting every vertex. We
consider the approximability of ATSP on topologically restricted graphs. It has
been shown by [Oveis Gharan and Saberi 2011] that there exists polynomial-time
constant-factor approximations on planar graphs and more generally graphs of
constant orientable genus. This result was extended to non-orientable genus by
[Erickson and Sidiropoulos 2014].
We show that for any class of \emph{nearly-embeddable} graphs, ATSP admits a
polynomial-time constant-factor approximation. More precisely, we show that for
any fixed k≥0, there exist α,β>0, such that ATSP on
n-vertex k-nearly-embeddable graphs admits a α-approximation in time
O(nβ). The class of k-nearly-embeddable graphs contains graphs with at
most k apices, k vortices of width at most k, and an underlying surface
of either orientable or non-orientable genus at most k. Prior to our work,
even the case of graphs with a single apex was open. Our algorithm combines
tools from rounding the Held-Karp LP via thin trees with dynamic programming.
We complement our upper bounds by showing that solving ATSP exactly on graphs
of pathwidth k (and hence on k-nearly embeddable graphs) requires time
nΩ(k), assuming the Exponential-Time Hypothesis (ETH). This is
surprising in light of the fact that both TSP on undirected graphs and Minimum
Cost Hamiltonian Cycle on directed graphs are FPT parameterized by treewidth