  We generalize the monotone local search approach of Fomin, Gaspers,
Lokshtanov and Saurabh [J.ACM 2019], by establishing a connection between
parameterized approximation and exponential-time approximation algorithms for
monotone subset minimization problems. In a monotone subset minimization
problem the input implicitly describes a non-empty set family over a universe
of size $n$ which is closed under taking supersets. The task is to find a
minimum cardinality set in this family. Broadly speaking, we use approximate
monotone local search to show that a parameterized $\alpha$-approximation
algorithm that runs in $c^k \cdot n^{O(1)}$ time, where $k$ is the solution
size, can be used to derive an $\alpha$-approximation randomized algorithm that
runs in $d^n \cdot n^{O(1)}$ time, where $d$ is the unique value in $d \in
(1,1+\frac{c-1}{\alpha})$ such that
$\mathcal{D}(\frac{1}{\alpha}\|\frac{d-1}{c-1})=\frac{\ln c}{\alpha}$ and
$\mathcal{D}(a \|b)$ is the Kullback-Leibler divergence. This running time
matches that of Fomin et al. for $\alpha=1$, and is strictly better when
$\alpha >1$, for any $c > 1$. Furthermore, we also show that this result can be
derandomized at the expense of a sub-exponential multiplicative factor in the
running time.
  We demonstrate the potential of approximate monotone local search by deriving
new and faster exponential approximation algorithms for Vertex Cover,
$3$-Hitting Set, Directed Feedback Vertex Set, Directed Subset Feedback Vertex
Set, Directed Odd Cycle Transversal and Undirected Multicut. For instance, we
get a $1.1$-approximation algorithm for Vertex Cover with running time $1.114^n
\cdot n^{O(1)}$, improving upon the previously best known $1.1$-approximation
running in time $1.127^n \cdot n^{O(1)}$ by Bourgeois et al. [DAM 2011].
