  A recent trend in parameterized algorithms is the application of polytope
tools (specifically, LP-branching) to FPT algorithms (e.g., Cygan et al., 2011;
Narayanaswamy et al., 2012). However, although interesting results have been
achieved, the methods require the underlying polytope to have very restrictive
properties (half-integrality and persistence), which are known only for few
problems (essentially Vertex Cover (Nemhauser and Trotter, 1975) and Node
Multiway Cut (Garg et al., 1994)). Taking a slightly different approach, we
view half-integrality as a \emph{discrete} relaxation of a problem, e.g., a
relaxation of the search space from $\{0,1\}^V$ to $\{0,1/2,1\}^V$ such that
the new problem admits a polynomial-time exact solution. Using tools from CSP
(in particular Thapper and \v{Z}ivn\'y, 2012) to study the existence of such
relaxations, we provide a much broader class of half-integral polytopes with
the required properties, unifying and extending previously known cases.
  In addition to the insight into problems with half-integral relaxations, our
results yield a range of new and improved FPT algorithms, including an
$O^*(|\Sigma|^{2k})$-time algorithm for node-deletion Unique Label Cover with
label set $\Sigma$ and an $O^*(4^k)$-time algorithm for Group Feedback Vertex
Set, including the setting where the group is only given by oracle access. All
these significantly improve on previous results. The latter result also implies
the first single-exponential time FPT algorithm for Subset Feedback Vertex Set,
answering an open question of Cygan et al. (2012).
  Additionally, we propose a network flow-based approach to solve some cases of
the relaxation problem. This gives the first linear-time FPT algorithm to
edge-deletion Unique Label Cover.
