From gap-exponential time hypothesis to fixed parameter tractable inapproximability: Clique, dominating set, and more

Parinya Chalermsook, Marek Cygan, Guy Kortsarz, Bundit Laekhanukit, Pasin Manurangsi, Danupon Nanongkai, Luca Trevisan

Research output: Contribution to journalArticlepeer-review


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 (Clique) and Minimum Dominating Set (DomSet) problems parameterized by the size of the optimal solution. In particular, letting OPT be the optimum and N be the size of the input, is there an algorithm that runs in t(OPT)poly(N) time and outputs a solution of size f(OPT) for any computable functions t and f that are independent of N (for Clique, we want f(OPT) = ω (1))? In this paper, we show that both Clique and DomSet admit no nontrivial FPT-approximation algorithm, i.e., there is no o(OPT)-FPT-approximation algorithm for Clique and no f(OPT)-FPT-approximation algorithm for DomSet for any function f. In fact, our results imply something even stronger: The best way to solve Clique and 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 2o(n)-time algorithm can distinguish between a satisfiable 3 SAT formula and one which is not even (1 - ε)-satisfiable for some constant ε > 0. Besides Clique and 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 ko(1)-FPT-approximation algorithm for the Densest k-Subgraph problem.

Original languageEnglish
Pages (from-to)772-810
Number of pages39
JournalSIAM Journal on Computing
Issue number4
StatePublished - 2020
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2020 Society for Industrial and Applied Mathematics


  • Clique
  • Dominating set
  • Fine-grained complexity
  • Hardness of approximation
  • Parameterized complexity
  • Subexponential-time algorithms


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