Linear-Time FPT Algorithms via Half-Integral Non-returning A-path Packing.

A recent trend in the design of FPT algorithms is exploiting half-integrality of LP relaxations. That is, 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 small 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 observing that, for many problems, the LP relaxation can be seen as the dual of the LP relaxation of a non-returning $A$-path packing problem. Then, we provide an $O(km)$-time algorithm that computes a half-integral optimal solution to the latter problem, where $k$ is the optimal value and $m$ is the number of edges. This improves the computational time even for a special case of the non-returning $A$-path packing, called internally-disjoint $A$-path packing. As a corollary, we obtain 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. For each of these problems, the obtained running time is linear in the input size and has the current smallest dependency on the parameter. In particular, these are the first linear-time FPT algorithms for Group Feedback Vertex Set and Non-Monochromatic Cycle Transversal.