Branch-width of connectivity functions is fixed-parameter tractable

TL;DR

This paper presents a fixed-parameter tractable algorithm for computing branch-decompositions of connectivity functions, significantly improving efficiency and resolving an open problem in the parameterized complexity of graph and matroid width parameters.

Contribution

It introduces an FPT algorithm for branch-width of connectivity functions given by an oracle, improving previous algorithms and solving an open problem in the field.

Findings

Algorithm runs in time $2^{O(k^2)} \, ext{poly}(n)$

Applicable to rank-width, branch-width of matroids, and carving-width

Improves dependency on $k$ in related FPT algorithms

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 $2^{O(k^2)} \gamma n^6 \log n$, where $\gamma$ 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 $\gamma n^{O(k)}$. 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 Hlin\v{e}n\'y [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.