Branch-width of connectivity functions is fixed-parameter tractable

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.

💡 Research Summary

The paper studies the branch‑width of connectivity functions, a broad class that includes graph rank‑width, graph carving‑width, graph branch‑width, and matroid branch‑width. A connectivity function f on a finite set V is an integer‑valued function that is symmetric ( f(X)=f(V∖X) ), submodular ( f(X)+f(Y)≥f(X∩Y)+f(X∪Y) ), and satisfies f(∅)=0. The authors prove that deciding whether bw(f)≤k and, if so, constructing a branch‑decomposition of width at most k is fixed‑parameter tractable (FPT) with respect to the parameter k. Their algorithm runs in time 2^{O(k²)}·γ·n⁶·log n, where γ is the time required to evaluate f on any subset and n=|V|. This improves the previous best algorithm by Oum and Seymour (2007), which required γ·n^{O(k)} time (an XP algorithm).

The central technical contribution is the notion of a “safe cut”. A cut (A,B) of V is safe if there exists an optimal‑width branch‑decomposition whose corresponding tree edge separates exactly A and B. The algorithm repeatedly finds safe cuts and recursively decomposes the problem on the two sides. The existence of a safe cut is guaranteed when the current instance is large relative to the target width: if a branch‑decomposition of width ℓ exists and 3ℓ+1<|V|, a safe cut with |A|,|B|≥2 can be found in 2^{O(ℓ)}·γ·n time.

To locate a safe cut the authors introduce “titanic” sets. A set X⊂V is titanic if for every tripartition (X₁,X₂,X₃) of X there is an index i with f(X_i)≥f(X). Testing whether a set is titanic reduces to a covering‑by‑three‑flats problem on a polymatroid derived from f. Using submodular function minimisation (the algorithm of Chakrabarty, Lee, Sidford, and Wong) they can either produce a violating tripartition or certify that the set is titanic, all within O(γ·3^{f(X)}·|X|) time.

The overall algorithm proceeds in three phases:

1. Initial Approximation – Using the Oum‑Seymour 3‑approximation (or iterative compression) they obtain a branch‑decomposition of width at most 3k or certify that bw(f)>k.

2. Recursive Safe‑Cut Extraction – While the instance size exceeds 2^{O(k)} they repeatedly find a safe cut using the titanic‑set test, split the instance into f_A and f_B, and recurse. Each recursion reduces the ground set size by at least a factor of two, so the depth is O(log n). The work per level is 2^{O(k)}·γ·|V|, leading to a total of 2^{O(k²)}·γ·n⁶·log n for this phase.

3. Base‑Case Exact Solution – When the remaining ground set has size at most 2^{O(k)} they invoke the Oum‑Seymour exact XP algorithm, which runs in 2^{O(k²)}·γ·n time, to obtain an optimal branch‑decomposition.

When the connectivity function comes from a matroid rank oracle, γ is the time to compute the rank of a set, and the same bound yields an FPT algorithm for matroid branch‑width. This resolves the open problem posed by Hliněný (2005) about the parameterised complexity of matroid branch‑width given by a rank oracle.

The algorithm also immediately applies to graph parameters: rank‑width, carving‑width, and ordinary branch‑width. For each of these, the previous best FPT algorithms had parameter dependencies of at least 2^{Ω(k³)} or even doubly‑exponential; the new method uniformly achieves a 2^{O(k²)} dependency, a substantial improvement.

The paper concludes with several directions for future work: (i) designing specialised implementations for restricted graph classes (planar, bounded‑treewidth, etc.) that could lower the constant in the exponent, (ii) exploiting newer, faster submodular minimisation techniques to improve practical performance, and (iii) finding purely combinatorial algorithms for the titanic‑set test that avoid the heavy submodular optimisation step.

In summary, the authors provide a conceptually simple yet powerful FPT framework for branch‑width of any connectivity function, delivering the first algorithm with running time 2^{O(k²)}·poly(n) and settling a long‑standing open question in matroid algorithmics.