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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, and [S.-I. Oum, Approximating rank-width and clique-width quickly, in Proceedings of the 31st International Workshop on Graph-Theoretic Concepts in Computer Science, Springer-Verlag, Heidelberg, to appear] a cubic time approximation algorithm for graph rank-width and clique-width.