On Parameterized Exponential Time Complexity

In this paper, we show that special instances of parameterized NP-hard problems are as difficult as the general instances in terms of their subexponential time computability. For example, we show that the Planar Dominating Set problem on degree-3 graphs can be solved in $2^{o(\sqrt{k})} p(n)$ parameterized time if and only if the general Planar Dominating Set problem can. Apart from their complexity theoretic implications, our results have some interesting algorithmic implications as well.

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