The Constant Inapproximability of the Parameterized Dominating Set Problem

Abstract

A set $D$ of vertices of a graph $G$ is a dominating set if every vertex of $G$ is contained in $D$ or adjacent to some vertex of $D$. The number of vertices in a smallest dominating set of $G$ is denoted by $\gamma(G)$. We prove that, under the assumption ${FPT}\ne {W}[1]$ from parameterized complexity, for any constant $c\in \mathbb{N}^+$ and computable function $f: \mathbb{N}\to \mathbb{N}$ there is no algorithm which on every input graph $G$ finds a dominating set of size at most $c\cdot \gamma(G)$ in $f(\gamma(G))\cdot |G|^{O(1)}$ time. In other words, any constant approximation of the parameterized dominating set problem is ${W}[1]$-hard. Furthermore, assuming the exponential time hypothesis (ETH) [R. Impagliazzo and R. Paturi, J. Comput. System Sci., 62 (2001), pp. 367--375], we can even rule out the existence of a $f(\gamma(G))\cdot |G|^{{({log}\;\gamma(G))}^{{\varepsilon}/{12}}}$-time algorithm which on every input graph $G$ outputs a dominating set of size at most $\sqrt[3+\varepsilon]{{log}\; (\gamma(G))} \cdot \gamma(G)$ for every $0<\varepsilon<1$. Our hardness reduction is built on the second author's recent ${W}[1]$-hardness proof of the biclique problem [B. Lin, The parameterized complexity of $k$-Biclique, in Proceedings of the 26th Annual ACM--SIAM Symposium on Discrete Algorithms, SODA 2015 (San Diego, CA), ACM, New York, SIAM, Philadelphia, 2015, pp. 605--615]. This yields, among other things, a proof without the probabilistically checkable proof (PCP) machinery that the classic dominating set problem has no polynomial time constant approximation under ETH.