  A recent trend in the design of FPT algorithms is exploiting the
half-integrality of LP relaxations. In other words, starting with a
half-integral optimal solution to an LP relaxation, we assign integral values
to variables one-by-one by branch and bound. This technique is general and the
resulting time complexity has a low dependency on the parameter. However, the
time complexity often becomes a large polynomial in the input size because we
need to compute half-integral optimal LP solutions.
  In this paper, we address this issue by providing an $O(km)$-time algorithm
for solving the LPs arising from various FPT problems, where $k$ is the optimal
value and $m$ is the number of edges/constraints. Our algorithm is based on
interesting connections among 0/1/all constraints, which has been studied in
the field of constraints satisfaction, $A$-path packing, which has been studied
in the field of combinatorial optimization, and the LPs used in FPT algorithms.
With the aid of this algorithm, we obtain improved FPT algorithms for various
problems, including Group Feedback Vertex Set, Subset Feedback Vertex Set, Node
Multiway Cut, Node Unique Label Cover, and Non-monochromatic Cycle Transversal.
The obtained running time for each of these problems is linear in the input
size and has the current smallest dependency on the parameter. In particular,
these algorithms are the first linear-time FPT algorithms for problems
including Group Feedback Vertex Set and Non-monochromatic Cycle Transversal.
