We consider questions that arise from the intersection between the areas of
polynomial-time approximation algorithms, subexponential-time algorithms, and
fixed-parameter tractable algorithms. The questions, which have been asked
several times (e.g., [Marx08, FGMS12, DF13]), are whether there is a
non-trivial 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 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 non-trivial
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
(e.g., this holds even if f is the Ackermann function). 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 (Gap-ETH) [Dinur16, MR16], which
states that no 2o(n)-time algorithm can distinguish between a satisfiable
3SAT formula and one which is not even (1−ϵ)-satisfiable for some
constant ϵ>0.
Besides Clique and DomSet, we also rule out non-trivial FPT-approximation for
Maximum Balanced Biclique, Maximum Subgraphs with Hereditary Properties, and
Maximum Induced Matching in bipartite graphs. Additionally, we rule out
ko(1)-FPT-approximation algorithm for Densest k-Subgraph although this
ratio does not yet match the trivial O(k)-approximation algorithm.Comment: 43 pages. To appear in FOCS'1