Finding branch-decompositions of matroids, hypergraphs, and more

Abstract

Given $n$ subspaces of a finite-dimensional vector space over a fixed finite field $\mathcal F$, 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 with leaves in one component of $T-e$ and the sum of subspaces associated with leaves in the other component have the intersection of dimension at most $k$. This problem includes the problems of computing branch-width of $\mathcal F$-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 $\mathcal F$. Our algorithm is analogous to the algorithm of Bodlaender and Kloks (1996) 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 $\mathcal F$-represented matroids was due to Hlin\v{e}n\'y and Oum (2008) that runs in time $O(n^3)$ where $n$ is the number of elements of the input $\mathcal F$-represented matroid. But their method is highly indirect. Their algorithm uses the non-trivial fact by Geelen et al. (2003) that the number of forbidden minors is finite and uses the algorithm of Hlin\v{e}n\'y (2005) on checking monadic second-order formulas on $\mathcal F$-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 $k$.Comment: 73 pages, 10 figure