Branch-width of connectivity functions is fixed-parameter tractable
Tuukka Korhonen, Sang-il Oum
TL;DR
This work proves that the branch-width of connectivity functions is fixed-parameter tractable: given a connectivity function $f$ on $n$ elements and an oracle time $\gamma$ to compute $f(X)$, one can either construct a branch-decomposition of width at most $k$ or certify that $bw(f)>k$ in time $2^{O(k^2)}\gamma n^6\log n$. The approach combines safe cuts, titanic subsets, and a divide-and-conquer strategy that reduces large instances to smaller ones, with an exact subroutine for the compactized case. The authors also adapt the method to matroids given by rank oracles, achieving improved runtimes via matroid intersection, and they explain implications for rank-width, carving-width, and related width parameters. This resolves an open problem by Hliněný and yields faster dependency on $k$ for several width-type parameters in graphs and matroids. Overall, the paper unifies and strengthens FPT results across multiple combinatorial width notions by introducing a structurally driven, scalable decomposition framework.
Abstract
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.