We give approximation schemes for Subset TSP and Steiner Tree on unit disk graphs, and more generally, on intersection graphs of similarly sized connected fat (not necessarily convex) polygons in the plane. As a first step towards this goal, we prove spanner-type results: finding an induced subgraph of bounded size that is $(1+\varepsilon)$-equivalent to the original instance in the sense that the optimum value increases only by a factor of at most $(1+\varepsilon)$ when the solution can use only the edges in this subgraph. - For Subset TSP, our algorithms find a $(1+\varepsilon)$-equivalent induced subgraph of size $\mathrm{poly}(1/\varepsilon)\cdot\mathrm{OPT}$ in polynomial time, and use it to find a $(1+\varepsilon)$-approximate solution in time $2^{\mathrm{poly}(1/\varepsilon)}\cdot n^{O(1)}$. - For Steiner Tree, our algorithms find a $(1+\varepsilon)$-equivalent induced subgraph of size $2^{\mathrm{poly}(1/\varepsilon)}\cdot\mathrm{OPT}$ in time $2^{\mathrm{poly}(1/\varepsilon)}\cdot n^{O(1)}$, and use it to find a $(1+\varepsilon)$-approximate solution in time $2^{2^{\mathrm{poly}(1/\varepsilon)}}\cdot n^{O(1)}$. - An improved algorithm finds a $(1+\varepsilon)$-approximate solution for Steiner Tree in time $2^{\mathrm{poly}(1/\varepsilon)}\cdot n^{O(1)}$. An easy reduction shows that approximation schemes for unit disks imply approximation schemes for planar graphs. Thus our results are far-reaching generalizations of analogous results of Klein [STOC'06] and Borradaile, Klein, and Mathieu [ACM TALG'09] for Subset TSP and Steiner Tree in planar graphs. We show that our results are best possible in the sense that dropping any of (i) similarly sized, (ii) connected, or (iii) fat makes both problems APX-hard.

翻译：本文给出了单位圆盘图以及更一般情形下平面上相似尺寸连通胖（不必为凸）多边形交图的子集旅行商问题与斯坦纳树的近似方案。作为实现该目标的第一步，我们证明了跨接型结果：存在一个规模有界的诱导子图，当解仅能使用该子图中的边时，最优值最多增加$(1+\varepsilon)$倍，即该子图与原实例$(1+\varepsilon)$等价。具体而言： - 针对子集TSP，我们的算法在多项式时间内找到一个规模为$\mathrm{poly}(1/\varepsilon)\cdot\mathrm{OPT}$的$(1+\varepsilon)$等价诱导子图，并利用该子图在$2^{\mathrm{poly}(1/\varepsilon)}\cdot n^{O(1)}$时间内求得$(1+\varepsilon)$近似解。 - 针对斯坦纳树，我们的算法在$2^{\mathrm{poly}(1/\varepsilon)}\cdot n^{O(1)}$时间内找到一个规模为$2^{\mathrm{poly}(1/\varepsilon)}\cdot\mathrm{OPT}$的$(1+\varepsilon)$等价诱导子图，并利用该子图在$2^{2^{\mathrm{poly}(1/\varepsilon)}}\cdot n^{O(1)}$时间内求得$(1+\varepsilon)$近似解。 - 改进算法可在$2^{\mathrm{poly}(1/\varepsilon)}\cdot n^{O(1)}$时间内直接求得斯坦纳树的$(1+\varepsilon)$近似解。 通过简单归约可知，单位圆盘的近似方案可推导出平面图的近似方案。因此，我们的结果是对Klein [STOC'06]及Borradaile、Klein与Mathieu [ACM TALG'09]关于平面图上子集TSP与斯坦纳树类似结果的高度泛化。我们证明，在（i）相似尺寸、（ii）连通性、（iii）胖性条件中任意缺失一条，两个问题均变为APX-hard，从而表明我们的结果在本质上已达到最优。