On the Parameterized Complexity of Approximating Dominating Set

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. 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. 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.