  In a recent work, Esmer et al. describe a simple method - Approximate
Monotone Local Search - to obtain exponential approximation algorithms from
existing parameterized exact algorithms, polynomial-time approximation
algorithms and, more generally, parameterized approximation algorithms. In this
work, we generalize those results to the weighted setting.
  More formally, we consider monotone subset minimization problems over a
weighted universe of size $n$ (e.g., Vertex Cover, $d$-Hitting Set and Feedback
Vertex Set). We consider a model where the algorithm is only given access to a
subroutine that finds a solution of weight at most $\alpha \cdot W$ (and of
arbitrary cardinality) in time $c^k \cdot n^{O(1)}$ where $W$ is the minimum
weight of a solution of cardinality at most $k$. In the unweighted setting,
Esmer et al. determine the smallest value $d$ for which a $\beta$-approximation
algorithm running in time $d^n \cdot n^{O(1)}$ can be obtained in this model.
We show that the same dependencies also hold in a weighted setting in this
model: for every fixed $\varepsilon>0$ we obtain a $\beta$-approximation
algorithm running in time $O\left((d+\varepsilon)^{n}\right)$, for the same $d$
as in the unweighted setting.
  Similarly, we also extend a $\beta$-approximate brute-force search (in a
model which only provides access to a membership oracle) to the weighted
setting. Using existing approximation algorithms and exact parameterized
algorithms for weighted problems, we obtain the first exponential-time
$\beta$-approximation algorithms that are better than brute force for a variety
of problems including Weighted Vertex Cover, Weighted $d$-Hitting Set, Weighted
Feedback Vertex Set and Weighted Multicut.
