Bounded Fixed-Parameter Tractability and log² n Nondeterministic Bits

Abstract

Motivated by recent results showing that there are natural parameterized problems that are fixed-parameter tractable, but can only be solved by fixed-parameter tractable algorithms the running time of which depends non-elementarily on the parameter, we propose a notion of bounded fixed-parameter tractability, where the dependence of the running time on the parameter is restricted to be singly exponential. We develop a basic theory that is centred around the class EPT of tractable problems and an EW-hierarchy of classes of intractable problems, both in the bounded sense. By and large, this theory is similar to the established unbounded parameterized complexity theory, but there are some remarkable differences. Most notably, certain natural model-checking problems that are known to be fixed-parameter tractable in the unbounded sense have a very high complexity in the bounded theory. The problem of computing the VC-dimension of a family of sets, which is known to be complete for the class W[1] in the unbounded theory, is complete for the class EW[3] in the bounded theory. It turns out that our bounded parameterized complexity theory is closely related to the classical complexity theory of problems that can be solved by a nondeterministic polynomial time algorithm that only uses log² n nondeterministic bits, and in particular to the classes LOGSNP and LOGNP introduced by Papadimitriou and Yannakakis [15].