A Parametrized Algorithm for Matroid Branch-Width

Branch-width is a structural parameter very closely related to tree-width, but branch-width has an immediate generalization from graphs to matroids. We present an algorithm that, for a given matroid M of bounded branch-width t which is represented over a finite field, finds a branch decomposition of M of width at most 3t in cubic time. Then we show that the branch-width of M is a uniformly fixed-parameter tractable problem. Other applications include recognition of matroid properties definable in the monadic second-order logic for bounded branch-width, or [Oum] a cubic approximation algorithm for graph rank-width and clique-width.