非对称旅行商问题（ATSP）的目标是在一个有向图中找到一个访问每个顶点的最小费用的闭行走。我们考虑ATSP在拓扑限制图上的可逼近性。[Oveis Gharan and Saberi 2011]已经证明了平面图和更一般的常定向亏格的图上存在多项式时间常数因子近似。这一结果由[Erickson and Sidiropoulos 2014]扩展到不可定向的属。我们证明了对于任何一类\emph{nearly-embeddable}图，ATSP都有多项式时间常数因子近似。更精确地说，我们证明了，对于任何固定的$k\geq 0$，存在$α, β>0$，使得ATSP上的$n$ -顶点$k$ -近嵌入图允许一个$α$ -近似的时间$O(n^β)$。一类$k$ -几乎可嵌入图包含至多有$k$顶点、至多有$k$宽度的$k$涡、至多有$k$可定向亏格或不可定向亏格的底层曲面的图.在我们的工作之前，即使是单顶点图的情况也是开放的。我们的算法结合了工具，从四舍五入的Held-Karp LP通过薄树与动态规划。我们补充我们的上界显示，解决ATSP的路径宽度$k$图（因此在$k$ -近嵌入图）需要时间$n^{Ω(k)}$，假设指数时间假设（ETH）。这是令人惊讶的，因为无向图上的TSP和有向图上的最小成本哈密尔顿圈都是由树宽参数化的FPT。
英文摘要：
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 $k\geq 0$, there exist $α, β>0$, such that ATSP on $n$-vertex $k$-nearly-embeddable graphs admits a $α$-approximation in time $O(n^β)$. 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 $n^{Ω(k)}$, 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.
