From Gap-Exponential Time Hypothesis to Fixed Parameter Tractable Inapproximability: Clique, Dominating Set, and More

Abstract

We consider questions that arise from the intersection between the areas of polynomial-time approximation algorithms, subexponential-time algorithms, and fixed-parameter tractable (FPT) algorithms. The questions, which have been asked several times, are whether there is a nontrivial FPT-approximation algorithm for the Maximum Clique $({\sf Clique})$ and Minimum Dominating Set $({\sf DomSet})$ problems parameterized by the size of the optimal solution. In particular, letting ${\sf OPT}$ be the optimum and $N$ be the size of the input, is there an algorithm that runs in $t({\sf OPT}){\operatorname{poly}}(N)$ time and outputs a solution of size $f({\sf OPT})$ for any computable functions $t$ and $f$ that are independent of $N$ (for ${\sf Clique}$, we want $f({\sf OPT})=\omega(1)$)? In this paper, we show that both ${\sf Clique}$ and ${\sf DomSet}$ admit no nontrivial FPT-approximation algorithm, i.e., there is no $o({\sf OPT})$-FPT-approximation algorithm for ${\sf Clique}$ and no $f({\sf OPT})$-FPT-approximation algorithm for ${\sf DomSet}$ for any function $f$. In fact, our results imply something even stronger: The best way to solve ${\sf Clique}$ and ${\sf DomSet}$, even approximately, is to essentially enumerate all possibilities. Our results hold under the Gap Exponential Time Hypothesis [I. Dinur. ECCC, TR16-128, 2016; P. Manurangsi and P. Raghavendra, preprint, arXiv:1607.02986, 2016], which states that no $2^{o(n)}$-time algorithm can distinguish between a satisfiable 3 \sf SAT formula and one which is not even $(1 - \varepsilon)$-satisfiable for some constant $\varepsilon > 0$. Besides ${\sf Clique}$ and ${\sf DomSet}$, we also rule out nontrivial FPT-approximation for the Maximum Biclique problem, the problem of finding maximum subgraphs with hereditary properties (e.g., Maximum Induced Planar Subgraph), and Maximum Induced Matching in bipartite graphs, and we rule out the $k^{o(1)}$-FPT-approximation algorithm for the Densest $k$-Subgraph problem.

Topics & Concepts