Finding Branch-Decompositions of Matroids, Hypergraphs, and More
Jisu Jeong, Eun Jung Kim, Sang‐il Oum · SIAM Journal on Discrete Mathematics · 2021
Given $n$ subspaces of a finite-dimensional vector space over a fixed finite field ${\mathbb F}$, we wish to find a “branch-decomposition” of these subspaces of width at most $k$ that is a subcubic tree $T$ with $n$ leaves mapped bijectively to the subspaces such that for every edge $e$ of $T$, the sum of subspaces associated to the leaves in one component of $T-e$ and the sum of subspaces associated to the leaves in the other component have the intersection of dimension at most $k$. This problem includes the problems of computing branch-width of ${\mathbb F}$-represented matroids, rank-width of graphs, branch-width of hypergraphs, and carving-width of graphs. We present a fixed-parameter algorithm to construct such a branch-decomposition of width at most $k$, if it exists, for input subspaces of a finite-dimensional vector space over ${\mathbb F}$. Our algorithm is analogous to the algorithm of Bodlaender and Kloks [ J. Algorithms, 21 (1996), pp. 358--402] on tree-width of graphs. To extend their framework to branch-decompositions of vector spaces, we developed highly generic tools for branch-decompositions on vector spaces. The only known previous fixed-parameter algorithm for branch-width of ${\mathbb F}$-represented matroids was due to Hliněný and Oum [ SIAM J. Comput., 38 (2008), pp. 1012--1032] that runs in time $O(n^3)$ where $n$ is the number of elements of the input ${\mathbb F}$-represented matroid. But their method is highly indirect. Their algorithm uses the nontrivial fact by Geelen et al. [ J. Combin. Theory Ser. B, 88 (2003), pp. 261--265] that the number of forbidden minors is finite and uses the algorithm of Hliněný [ J. Combin. Theory Ser. B, 96 (2006), pp. 325--351] on checking monadic second-order formulas on ${\mathbb F}$-represented matroids of small branch-width. Our result does not depend on such a fact and is completely self-contained, and yet matches their asymptotic running time for each fixed $k$.