  In this paper we design {\sf FPT}-algorithms for two parameterized problems.
The first is \textsc{List Digraph Homomorphism}: given two digraphs $G$ and $H$
and a list of allowed vertices of $H$ for every vertex of $G$, the question is
whether there exists a homomorphism from $G$ to $H$ respecting the list
constraints. The second problem is a variant of \textsc{Multiway Cut}, namely
\textsc{Min-Max Multiway Cut}: given a graph $G$, a non-negative integer
$\ell$, and a set $T$ of $r$ terminals, the question is whether we can
partition the vertices of $G$ into $r$ parts such that (a) each part contains
one terminal and (b) there are at most $\ell$ edges with only one endpoint in
this part. We parameterize \textsc{List Digraph Homomorphism} by the number $w$
of edges of $G$ that are mapped to non-loop edges of $H$ and we give a time
$2^{O(\ell\cdot\log h+\ell^2\cdot \log \ell)}\cdot n^{4}\cdot \log n$
algorithm, where $h$ is the order of the host graph $H$. We also prove that
\textsc{Min-Max Multiway Cut} can be solved in time $2^{O((\ell r)^2\log \ell
r)}\cdot n^{4}\cdot \log n$. Our approach introduces a general problem, called
{\sc List Allocation}, whose expressive power permits the design of
parameterized reductions of both aforementioned problems to it. Then our
results are based on an {\sf FPT}-algorithm for the {\sc List Allocation}
problem that is designed using a suitable adaptation of the {\em randomized
contractions} technique (introduced by [Chitnis, Cygan, Hajiaghayi, Pilipczuk,
and Pilipczuk, FOCS 2012]).
