A connectivity function on a finite set $V$ is a symmetric submodular function $f \colon 2^V \to \mathbb{Z}$ with $f(\emptyset)=0$. 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 $2^{O(k^2)} γn^6 \log n$, where $γ$ is the time to compute $f(X)$ for any set $X$, and $n = |V|$. This improves the previous algorithm by Oum and Seymour [J. Combin. Theory Ser. B, 2007], which runs in time $γn^{O(k)}$. 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.

翻译：在有限集合$V$上的连通性函数是指满足$f(\emptyset)=0$的对称子模函数$f \colon 2^V \to \mathbb{Z}$。我们证明，对于由预言机给出的连通性函数，寻找宽度至多为$k$的枝分解是固定参数可处理（FPT）问题，并给出了运行时间为$2^{O(k^2)} γn^6 \log n$的算法，其中$γ$是计算任意集合$X$的函数值$f(X)$所需的时间，$n = |V|$。这改进了Oum和Seymour [J. Combin. Theory Ser. B, 2007] 提出的运行时间为$γn^{O(k)}$的算法。我们的算法可应用于图的秩宽、拟阵的枝宽、（超）图的枝宽以及图的雕刻宽。这解决了Hliněný [SIAM J. Comput., 2005] 提出的一个开放问题，即由秩预言机给出的拟阵枝宽是否为固定参数可处理问题。此外，我们的算法在图枝宽、秩宽和雕刻宽的FPT算法运行时间中，对参数$k$的依赖关系达到了目前已知的最佳改进。