Branch-width of connectivity functions is fixed-parameter tractable
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866911433428566016 |
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| author | Korhonen, Tuukka Oum, Sang-il |
| author_facet | Korhonen, Tuukka Oum, Sang-il |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_04756 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Branch-width of connectivity functions is fixed-parameter tractable Korhonen, Tuukka Oum, Sang-il Data Structures and Algorithms Discrete Mathematics Combinatorics 05C85, 05B35, 05C75 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. |
| title | Branch-width of connectivity functions is fixed-parameter tractable |
| topic | Data Structures and Algorithms Discrete Mathematics Combinatorics 05C85, 05B35, 05C75 |
| url | https://arxiv.org/abs/2601.04756 |