  The lower and the upper irredundance numbers of a graph $G$, denoted $ir(G)$
and $IR(G)$ respectively, are conceptually linked to domination and
independence numbers and have numerous relations to other graph parameters. It
is a long-standing open question whether determining these numbers for a graph
$G$ on $n$ vertices admits exact algorithms running in time less than the
trivial $\Omega(2^n)$ enumeration barrier. We solve these open problems by
devising parameterized algorithms for the dual of the natural parameterizations
of the problems with running times faster than $O^*(4^{k})$. For example, we
present an algorithm running in time $O^*(3.069^{k})$ for determining whether
$IR(G)$ is at least $n-k$. Although the corresponding problem has been known to
be in FPT by kernelization techniques, this paper offers the first
parameterized algorithms with an exponential dependency on the parameter in the
running time. Additionally, our work also appears to be the first example of a
parameterized approach leading to a solution to a problem in exponential time
algorithmics where the natural interpretation as an exact exponential-time
algorithm fails.

Data Structures and Algorithms