Paper Info
Title
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 (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) = omega(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 2(o(n))-time algorithm can distinguish between a satisfiable 3 SAT formula and one which is not even (1 - epsilon)-satisfiable for some constant epsilon > 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 k(o(1))-FPT-approximation algorithm for the Densest k-Subgraph problem.
Year
DOI
Venue
2020
10.1137/18M1166869
SIAM JOURNAL ON COMPUTING
Keywords
DocType
Volume
hardness of approximation,parameterized complexity,subexponential-time algorithms,fine-grained complexity,clique,dominating set
Journal
49
Issue
ISSN
Citations 
4
0097-5397
0
PageRank 
References 
Authors
0.34
0
7
Name
Order
Citations
PageRank
Parinya Chalermsook118720.94
Marek Cygan255640.52
Guy Kortsarz31539126.16
Bundit Laekhanukit412916.93
Pasin Manurangsi56228.79
Danupon Nanongkai664341.26
Luca Trevisan72995232.34