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. As an immediate consequence, this yields a $1.39$-approximation polynomial time algorithm for CST in $\ell_p$-metrics.

翻译：在连续斯坦纳树问题（CST）中，给定度量空间中的一组点（称为终端点）作为输入，要求寻找连接它们的最小代价树。解中可引入度量空间中的额外点（称为斯坦纳点）作为节点。在离散斯坦纳树问题（DST）中，除终端点外，还需给定一组设施，且任何连接终端点的解树只能包含来自该设施集合的斯坦纳点。Trevisan [SICOMP'00] 证明，当输入位于 $\ell_1$ 度量（以及汉明度量）中时，CST 与 DST 是 APX 难的。Chlebík 和 Chlebíková [TCS'08] 证明，在图度量（进而 $\ell_\infty$ 度量）中，DST 的近似因子为 $96/95\approx 1.01$ 是 NP 难的。在此工作之前，尚不清楚 CST 与 DST 在几乎所有其他常见度量中是否为 APX 难。本文证明：DST 在每个 $\ell_p$ 度量中均为 APX 难；CST 在 $\ell_{\infty}$ 度量中为 APX 难。最后，我们建立 CST 与 DST 的关联，给出从 CST 到 $\ell_p$ 度量中 DST 的一般性归约。作为直接推论，该归约为 $\ell_p$ 度量中的 CST 问题提供了 $1.39$ 近似比的多项式时间算法。