In the Continuous Steiner Tree problem (CST), we are given as input a set of points (called terminals) in a metric space and ask for the minimum-cost tree connecting them. Additional points (called Steiner points) from the metric space can be introduced as nodes in the solution. In the Discrete Steiner Tree problem (DST), we are given in addition to the terminals, a set of facilities, and any solution tree connecting the terminals can only contain the Steiner points from this set of facilities. Trevisan [SICOMP'00] showed that CST and DST are APX-hard when the input lies in the $\ell_1$-metric (and Hamming metric). Chleb\'ik and Chleb\'ikov\'a [TCS'08] showed that DST is NP-hard to approximate to factor of $96/95\approx 1.01$ in the graph metric (and consequently $\ell_\infty$-metric). Prior to this work, it was unclear if CST and DST are APX-hard in essentially every other popular metric. In this work, we prove that DST is APX-hard in every $\ell_p$-metric. We also prove that CST is APX-hard in the $\ell_{\infty}$-metric. Finally, we relate CST and DST, showing a general reduction from CST to DST in $\ell_p$-metrics.

翻译：在连续Steiner树问题（CST）中，我们被给定度量空间中的一组点（称为终端点）作为输入，并要求找到连接这些点的最小代价树。解中允许引入度量空间中的额外点（称为Steiner点）作为节点。在离散Steiner树问题（DST）中，除终端点外我们还被给定一组设施点，且任何连接终端点的解树只能包含来自该设施点集的Steiner点。Trevisan [SICOMP'00] 证明了当输入位于$\ell_1$度量（及汉明度量）时，CST和DST是APX难解的。Chlebík与Chlebíková [TCS'08] 证明了在图度量（进而$\ell_\infty$度量）中，DST不存在优于$96/95\approx 1.01$近似因子的多项式时间算法（除非P=NP）。在本研究之前，尚不清楚CST和DST是否在几乎所有其他常用度量中都是APX难解的。本工作中，我们证明了DST在所有$\ell_p$度量中都是APX难解的。同时证明了CST在$\ell_{\infty}$度量中是APX难解的。最后，我们建立了CST与DST之间的联系，给出了$\ell_p$度量中从CST到DST的通用归约方法。