We study the parameterized complexity of approximating the k -Dominating Set (DomSet) problem where an integer k and a graph G on n vertices are given as input, and the goal is to find a dominating set of size at most F ( k ) ⋅ k whenever the graph G has a dominating set of size k . When such an algorithm runs in time T ( k ) ⋅ poly ( n ) (i.e., FPT-time) for some computable function T , it is said to be an F ( k )- FPT-approximation algorithm for k -DomSet. Whether such an algorithm exists is listed in the seminal book of Downey and Fellows (2013) as one of the “most infamous” open problems in parameterized complexity. This work gives an almost complete answer to this question by showing the non-existence of such an algorithm under W[1] ≠ FPT and further providing tighter running time lower bounds under stronger hypotheses. Specifically, we prove the following for every computable functions T , F and every constant ε > 0: • Assuming W[1] ≠ FPT, there is no F ( k )- FPT-approximation algorithm for k -DomSet. • Assuming the Exponential Time Hypothesis (ETH), there is no F ( k )-approximation algorithm for k -DomSet that runs in T ( k ) ⋅ n o ( k ) time. • Assuming the Strong Exponential Time Hypothesis (SETH), for every integer k ≥ 2, there is no F ( k )-approximation algorithm for k -DomSet that runs in T ( k ) ⋅ n k − ε time. • Assuming the k -SUM Hypothesis, for every integer k ≥ 3, there is no F ( k )-approximation algorithm for k -DomSet that runs in T ( k ) ⋅ n ⌈ k /2 ⌉ − ε time. Previously, only constant ratio FPT-approximation algorithms were ruled out under sf W[1] ≠ FPT and (log 1/4 &minus ε k )-FPT-approximation algorithms were ruled out under ETH [Chen and Lin, FOCS 2016]. Recently, the non-existence of an F ( k )-FPT-approximation algorithm for any function F was shown under Gap-ETH [Chalermsook et al., FOCS 2017]. Note that, to the best of our knowledge, no running time lower bound of the form n &delta k for any absolute constant δ > 0 was known before even for any constant factor inapproximation ratio. Our results are obtained by establishing a connection between communication complexity and hardness of approximation, generalizing the ideas from a recent breakthrough work of Abboud et al. [FOCS 2017]. Specifically, we show that to prove hardness of approximation of a certain parameterized variant of the label cover problem, it suffices to devise a specific protocol for a communication problem that depends on which hypothesis we rely on. Each of these communication problems turns out to be either a well-studied problem or a variant of one; this allows us to easily apply known techniques to solve them.

中文翻译：

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关于逼近支配集的参数化复杂度

我们研究了近似的参数化复杂度ķ- 支配集（DomSet）问题，其中一个整数ķ和一张图G在n顶点作为输入，目标是最多找到一个占主导地位的集合F(ķ) ⋅ķ每当图G有一组占主导地位的尺寸ķ. 当这样的算法及时运行吨(ķ) ⋅ 聚 (n) （即 FPT 时间）对于一些可计算的函数吨，据说是一个F(ķ)-FPT近似算法为了ķ-DomSet。Downey and Fellows (2013) 的开创性著作中将这种算法是否存在列为参数化复杂性中“最臭名昭著”的开放问题之一。这项工作通过显示在 W[1] ≠ FPT 下不存在这种算法，并在更强的假设下进一步提供更严格的运行时间下限，为这个问题提供了几乎完整的答案。具体来说，我们为每个可计算函数证明以下内容吨,F并且每个常数 ε > 0： • 假设 W[1] ≠ FPT，则不存在F(ķ)-FPT近似算法为了ķ-DomSet。• 假设指数时间假设（ETH），没有F(ķ)-近似算法ķ-DomSet 运行在吨(ķ) ⋅n ○(ķ)时间。• 假设强指数时间假设 (SETH)，对于每个整数ķ≥2，没有F(ķ)-近似算法ķ-DomSet 运行在吨(ķ) ⋅n ķ- ε时间。• 假设ķ-SUM 假设，对于每个整数ķ≥3，无F(ķ)-近似算法ķ-DomSet 运行在吨(ķ) ⋅n ⌈ķ/2 ⌉ - ε时间。以前，在 sf W[1] ≠ FPT 和 (log1/4&减 εķ)-FPT 近似算法在 ETH 下被排除 [Chen and Lin, FOCS 2016]。最近，一个不存在的F(ķ)-FPT-任意函数的逼近算法F显示在 Gap-ETH [Chalermsook 等人，FOCS 2017] 下。请注意，据我们所知，表格没有运行时间下限n ＆三角洲ķ 对于任何绝对常数 δ > 0，甚至对于任何常数因子近似比都是已知的。我们的结果是通过建立通信复杂性和近似硬度之间的联系获得的，概括了 Abboud 等人最近突破性工作的想法。[FOCS 2017]。具体来说，我们表明，为了证明标签覆盖问题的某个参数化变体的近似难度，只需为取决于我们所依赖的假设的通信问题设计一个特定的协议就足够了。这些沟通问题中的每一个都是经过充分研究的问题或问题的变体。这使我们能够轻松地应用已知技术来解决它们。