  We give a combinatorial condition for the existence of efficient, LP-based
FPT algorithms for a broad class of graph-theoretical optimisation problems.
Our condition is based on the notion of biased graphs known from matroid
theory. Specifically, we show that given a biased graph $\Psi=(G,\mathcal{B})$,
where $\mathcal{B}$ is a class of balanced cycles in $G$, the problem of
finding a set $X$ of at most $k$ vertices in $G$ which intersects every
unbalanced cycle in $G$ admits an FPT algorithm using an LP-branching approach,
similar to those previously seen for VCSP problems (Wahlstr\"om, SODA 2014).
This framework captures many of the problems previously solved via the VCSP
approach to LP-branching, as well as new generalisations, such as Group
Feedback Vertex Set for infinite groups (e.g., for graphs whose edges are
labelled by matrices). A major advantage compared to previous work is that it
is immediate to check the applicability of the result for a given problem,
whereas testing applicability of the VCSP approach for a specific VCSP requires
determining the existence of an embedding language with certain algebraically
defined properties, which is not known to be decidable in general.
Additionally, we study the approximation question, and show that every problem
of this category admits an $O(\log \text{OPT})$-approximation.

Data Structures and Algorithms