Subexponential Fixed-Parameter Tractability

Abstract

This last chapter of the book is concerned with subexponential fixed-parameter tractability, that is, with the class $\text{SUBEPT} = 2^{o^{\text{eff}}(k)}-\text{FPT}$. Subexponential fixed-parameter tractability is intimately linked with the theory of exact (exponential) algorithms for hard problems, which is concerned with algorithms for NP-hard problems that are better than the trivial exhaustive search algorithms, though still exponential. For example, there has been a long sequence of papers on exact algorithms for the 3-satisfiability problem; the currently best (randomized) algorithm for this problem has a running time of $1.324^n$ time for instances with n variables. There are numerous further examples of very nice nontrivial algorithms for hard problems, but a systematic complexity theory is still in its infancy. A question that has turned out to be central for such a theory is whether the 3-satisfiability problem can be solved in time $2^{o(n)}$.