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
ℓ, 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 ℓ 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
2O(ℓ⋅logh+ℓ2⋅logℓ)⋅n4⋅logn
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 2O((ℓr)2logℓr)⋅n4⋅logn. 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]).Comment: An extended abstract of this work will appear in the Proceedings of
the 10th International Symposium on Parameterized and Exact Computation
(IPEC), Patras, Greece, September 201