The D irected S teiner N etwork (DSN) problem takes as input a directed graph G =( V , E ) with non-negative edge-weights and a set D ⊆ V × V of k demand pairs. The aim is to compute the cheapest network N⊆ G for which there is an s\rightarrow t path for each ( s , t )∈ D. It is known that this problem is notoriously hard, as there is no k 1/4− o (1) -approximation algorithm under Gap-ETH, even when parametrizing the runtime by k [Dinur & Manurangsi, ITCS 2018]. In light of this, we systematically study several special cases of DSN and determine their parameterized approximability for the parameter k . For the bi -DSNP lanar problem, the aim is to compute a solution N⊆ G whose cost is at most that of an optimum planar solution in a bidirected graph G , i.e., for every edge uv of G the reverse edge vu exists and has the same weight. This problem is a generalization of several well-studied special cases. Our main result is that this problem admits a parameterized approximation scheme (PAS) for k . We also prove that our result is tight in the sense that (a) the runtime of our PAS cannot be significantly improved, and (b) no PAS exists for any generalization of bi-DSNP lanar , under standard complexity assumptions. The techniques we use also imply a polynomial-sized approximate kernelization scheme (PSAKS). Additionally, we study several generalizations of bi -DSNP lanar and obtain upper and lower bounds on obtainable runtimes parameterized by k . One important special case of DSN is the S trongly C onnected S teiner S ubgraph (SCSS) problem, for which the solution network N⊆ G needs to strongly connect a given set of k terminals. It has been observed before that for SCSS a parameterized 2-approximation exists for parameter k [Chitnis et al., IPEC 2013]. We give a tight inapproximability result by showing that for k no parameterized (2 − ε)-approximation algorithm exists under Gap-ETH. Additionally, we show that when restricting the input of SCSS to bidirected graphs, the problem remains NP-hard but becomes FPT for k .

中文翻译：

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双向施泰纳网络问题的参数化逼近算法

D被引导小号泰纳ñ网络(DSN) 问题将有向图作为输入G=(五,乙) 具有非负边缘权重和一组D⊆五×五的ķ需求对。目的是计算最便宜的网络N⊆G有一个s\右箭头 t每个路径（s,吨)∈D。众所周知，这个问题非常难，因为没有ķ 1/4−○(1)- Gap-ETH 下的近似算法，即使在通过参数化运行时ķ[Dinur & Manurangsi，ITCS 2018]。有鉴于此，我们系统地研究了 DSN 的几个特殊情况，并确定了它们对参数的参数化逼近性。ķ. 为了双-DSNP拉纳尔问题，目的是计算一个解决方案N⊆G其代价至多是双向图中最优平面解的代价G，即，对于每条边紫外线的G反向边缘似曾相识存在并且具有相同的权重。这个问题是几个经过充分研究的特殊情况的概括。我们的主要结果是这个问题允许参数化近似方案（PAS）ķ. 我们还证明了我们的结果是严格的，因为（a）我们的 PAS 的运行时间不能显着提高，并且（b）对于 bi-DSNP 的任何泛化都不存在 PAS拉纳尔，在标准复杂性假设下。我们使用的技术还暗示了多项式大小的近似核化方案 (PSAKS)。此外，我们研究了双-DSNP拉纳尔并获得参数化的可获取运行时的上限和下限ķ. DSN 的一个重要特例是 S强烈地C连接的小号泰纳小号子图（SCSS）问题，解决网络N⊆G需要强连接给定的一组ķ终端。之前已经观察到，对于 SCSS，参数存在参数化的 2 近似值ķ[Chitnis 等人，IPEC 2013]。我们通过证明对于ķGap-ETH 下不存在参数化 (2 − ε) 逼近算法。此外，我们表明，当将 SCSS 的输入限制为双向图时，问题仍然是 NP-hard，但对于ķ.