SIAM Journal on Computing, Volume 49, Issue 2, Page 318-364, January 2020.
Given a vertex-weighted directed graph $G=(V,E)$ and a set $T=\{t_1, t_2, \ldots, t_k\}$ of $k$ terminals, the objective of the Strongly Connected Steiner Subgraph (SCSS) problem is to find a vertex set $H\subseteq V$ of minimum weight such that $G[H]$ contains a $t_{i}\rightarrow t_j$ path for each $i\neq j$. The problem is NP-hard, but Feldman and Ruhl [SIAM J. Comput., 36 (2006), pp. 543--561] gave a novel $n^{O(k)}$ algorithm for the SCSS problem, where $n$ is the number of vertices in the graph and $k$ is the number of terminals. We explore how much easier the problem becomes on planar directed graphs. Our main algorithmic result is a $2^{O(k)}\cdot n^{O(\sqrt{k})}$ algorithm for planar SCSS, which is an improvement of a factor of $O(\sqrt{k})$ in the exponent over the algorithm of Feldman and Ruhl. Our main hardness result is a matching lower bound for our algorithm: we show that planar SCSS does not have an $f(k)\cdot n^{o(\sqrt{k})}$ algorithm for any computable function $f$, unless the exponential time hypothesis (ETH) fails. To obtain our algorithm, we first show combinatorially that there is a minimal solution whose treewidth is $O(\sqrt{k})$, and then use the dynamic-programming based algorithm for finding bounded-treewidth solutions due to Feldmann and Marx [The Complexity Landscape of Fixed-Parameter Directed Steiner Network Problems, preprint, ŭlhttps://arxiv.org/abs/1707.06808]. To obtain the lower bound matching the algorithm, we need a delicate construction of gadgets arranged in a gridlike fashion to tightly control the number of terminals in the created instance. The following additional results put our upper and lower bounds in context: our $2^{O(k)}\cdot n^{O(\sqrt{k})}$ algorithm for planar directed graphs can be generalized to graphs excluding a fixed minor. Additionally, we can obtain this running time for the problem of finding an optimal planar solution even if the input graph is not planar. In general graphs, we cannot hope for such a dramatic improvement over the $n^{O(k)}$ algorithm of Feldman and Ruhl: assuming ETH, SCSS in general graphs does not have an $f(k)\cdot n^{o(k/\log k)}$ algorithm for any computable function $f$. Feldman and Ruhl generalized their $n^{O(k)}$ algorithm to the more general Directed Steiner Network (DSN) problem; here the task is to find a subgraph of minimum weight such that for every source $s_i$ there is a path to the corresponding terminal $t_i$. We show that, assuming ETH, there is no $f(k)\cdot n^{o(k)}$ time algorithm for DSN on acyclic planar graphs. All our lower bounds hold for the integer weighted edge version, while the algorithm works for the more general unweighted vertex version.

中文翻译：

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具有固定数量的端子（和扩展）的平面强连通Steiner子图的紧界

SIAM计算杂志，第49卷，第2期，第318-364页，2020年1月。
给定顶点加权有向图$ G =（V，E）$和$ k $终端的集合$ T = \ {t_1，t_2，\ ldots，t_k \} $，强连接Steiner子图的目标（ SCSS）问题是找到一个最小权重的顶点集$ H \ subseteq V $，使得$ G [H] $包含每个$ i \ neq j $的$ t_ {i} \ rightarrow t_j $路径。这个问题很难解决，但是Feldman和Ruhl [SIAM J. Comput。，36（2006），pp.543--561]为SCSS问题提供了一种新颖的$ n ^ {O（k）} $算法，其中$ n $是图形中的顶点数，$ k $是端子数。我们探索问题在平面有向图上变得容易多了。我们的主要算法结果是平面SCSS的$ 2 ^ {O（k）} \ cdot n ^ {O（\ sqrt {k}）} $算法，这是对$ O（\ sqrt {k}的改进）$在Feldman和Ruhl算法上的指数。我们的主要硬度结果是算法的匹配下限：我们证明平面SCSS对于任何可计算函数$ f $都没有$ f（k）\ cdot n ^ {o（\ sqrt {k}）} $算法，除非指数时间假设（ETH）失败。为了获得我们的算法，我们首先组合显示一个最小解，其树宽为$ O（\ sqrt {k}）$，然后使用基于动态编程的算法来查找Feldmann和Marx [固定参数定向Steiner网络问题的复杂性，预印本，ŭlhttps：//arxiv.org/abs/1707.06808]。为了获得与算法匹配的下限，我们需要以网格状方式排列的小工具的精细构造，以紧密控制所创建实例中终端的数量。下面的其他结果将上下限置于上下文中：平面有向图的$ 2 ^ {O（k）} \ cdot n ^ {O（\ sqrt {k}）} $算法可以推广到不包括固定图的图上次要。此外，即使输入图不是平面的，我们也可以找到寻找最佳平面解的问题获得运行时间。在一般图形中，我们不能希望对Feldman和Ruhl的$ ​​n ^ {O（k）} $算法有如此大的改进：假设普通图形中的ETH，SCSS没有$ f（k）\ cdot n ^任何可计算函数$ f $的{o（k / \ log k）} $算法。Feldman和Ruhl将他们的$ n ^ {O（k）} $算法推广到更一般的定向斯坦纳网络（DSN）问题。此处的任务是找到最小权重的子图，以使每个源$ s_i $都有到相应终端$ t_i $的路径。我们证明 假设使用ETH，在无环平面图上没有DSN的$ f（k）\ cdot n ^ {o（k）} $时间算法。我们所有的下限都适用于整数加权边版本，而该算法适用于更通用的非加权顶点版本。