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，我们以度量空间中一组点（称为终端点）作为输入，并寻求连接它们的最小成本树。作为解中的节点，可以引入该度量空间中的额外点（称为斯坦纳点）。在离散斯坦纳树问题（DST）中，除终端点外，还给定一组设施，且任何连接终端点的解树仅能包含来自这组设施中的斯坦纳点。Trevisan [SICOMP'00] 指出，当输入位于 $\ell_1$ 度量（及汉明度量）中时，CST 和 DST 是 APX-难的。Chlebík 和 Chlebíková [TCS'08] 证明，在图度量（进而 $\ell_\infty$ 度量）中，DST 是 NP-难的，其近似比因子难以达到 $96/95\approx 1.01$。在此工作之前，尚不清楚 CST 和 DST 在几乎所有其他流行度量中是否为 APX-难的！本工作中，我们证明 DST 在每个 $\ell_p$ 度量中都是 APX-难的。我们还证明 CST 在 $\ell_{\infty}$ 度量中是 APX-难的。最后，我们关联 CST 与 DST，给出从 CST 到 $\ell_p$ 度量中 DST 的一般性归约。作为直接推论，这为 $\ell_p$ 度量中的 CST 问题提供了一个 $1.39$ 近似度的多项式时间算法。