The S TEINER T REE problem is one of the most fundamental NP-complete problems, as it models many network design problems. Recall that an instance of this problem consists of a graph with edge weights and a subset of vertices (often called terminals); the goal is to find a subtree of the graph of minimum total weight that connects all terminals. A seminal paper by Erickson et al. [Math. Oper. Res., 1987{ considers instances where the underlying graph is planar and all terminals can be covered by the boundary of k faces. Erickson et al. show that the problem can be solved by an algorithm using n O(k) time and n O(k) space, where n denotes the number of vertices of the input graph. In the past 30 years there has been no significant improvement of this algorithm, despite several efforts. In this work, we give an algorithm for P LANAR S TEINER T REE with running time 2 O(k) n O(√k) with the above parameterization, using only polynomial space. Furthermore, we show that the running time of our algorithm is almost tight: We prove that there is no f ( k ) n o(√k) algorithm for P LANAR S TEINER T REE for any computable function f , unless the Exponential Time Hypothesis fails.

中文翻译：

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平面 Steiner 树的近 ETH-tight 算法，在少数面上具有终端

小号泰纳吨稀土元素问题是最基本的 NP 完全问题之一，因为它模拟了许多网络设计问题。回想一下，这个问题的一个实例由一个带有边权重的图和一个顶点子集（通常称为终端）组成；目标是找到连接所有终端的最小总重量图的子树。Erickson 等人的开创性论文。[数学。操作。Res., 1987{ 考虑了底层图是平面的并且所有终端都可以被边界覆盖的情况ķ面孔。埃里克森等人。表明该问题可以通过使用以下算法解决n 好的）时间和n 好的）空间，在哪里n表示输入图的顶点数。在过去的 30 年中，尽管进行了几次努力，但该算法并没有显着改进。在这项工作中，我们给出了 P 的算法局域网小号泰纳吨稀土元素运行时间 2好的） n O(√k)通过上述参数化，仅使用多项式空间。此外，我们证明了我们算法的运行时间几乎是紧张的：我们证明没有F(ķ)n o(√k)P 的算法局域网小号泰纳吨稀土元素对于任何可计算函数F，除非指数时间假设失败。