Branch-width of connectivity functions is fixed-parameter tractable

A connectivity function on a finite set $V$ is a symmetric submodular function $f\text{\hspace{0.17em}⁣}:2^V\to \mathbb{Z}$ with $f(\mathrm{\varnothing })=0$. We prove that finding a branch-decomposition of width at most $k$ for a connectivity function given by an oracle is fixed-parameter tractable (FPT), by providing an algorithm of running time $2^{O(k^2)}\gamma n^6\log n$, where $\gamma$ is the time to compute $f(X)$ for any set $X$, and $n=\mathrm{\mid }V\mathrm{\mid }$. This improves the previous algorithm by Oum and Seymour [J. Combin. Theory Ser. B, 2007], which runs in time $\gamma n^{O(k)}$. Our algorithm can be applied to rank-width of graphs, branch-width of matroids, branch-width of (hyper)graphs, and carving-width of graphs. This resolves an open problem asked by Hliněný [SIAM J. Comput., 2005], who asked whether branch-width of matroids given by the rank oracle is fixed-parameter tractable. Furthermore, our algorithm improves the best known dependency on $k$ in the running times of FPT algorithms for graph branch-width, rank-width, and carving-width.