  We present two new combinatorial tools for the design of parameterized
algorithms. The first is a simple linear time randomized algorithm that given
as input a $d$-degenerate graph $G$ and an integer $k$, outputs an independent
set $Y$, such that for every independent set $X$ in $G$ of size at most $k$,
the probability that $X$ is a subset of $Y$ is at least $\left({(d+1)k \choose
k} \cdot k(d+1)\right)^{-1}$.The second is a new (deterministic) polynomial
time graph sparsification procedure that given a graph $G$, a set $T = \{\{s_1,
t_1\}, \{s_2, t_2\}, \ldots, \{s_\ell, t_\ell\}\}$ of terminal pairs and an
integer $k$, returns an induced subgraph $G^\star$ of $G$ that maintains all
the inclusion minimal multicuts of $G$ of size at most $k$, and does not
contain any $(k+2)$-vertex connected set of size $2^{{\cal O}(k)}$. In
particular, $G^\star$ excludes a clique of size $2^{{\cal O}(k)}$ as a
topological minor. Put together, our new tools yield new randomized fixed
parameter tractable (FPT) algorithms for Stable $s$-$t$ Separator, Stable Odd
Cycle Transversal and Stable Multicut on general graphs, and for Stable
Directed Feedback Vertex Set on $d$-degenerate graphs, resolving two problems
left open by Marx et al. [ACM Transactions on Algorithms, 2013]. All of our
algorithms can be derandomized at the cost of a small overhead in the running
time.
