Finding branch-decompositions of matroids, hypergraphs, and more
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Given n subspaces of a finite-dimensional vector space over a fixed finite field 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 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 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 F-represented matroids was due to Hliněný and Oum (2008) that runs in time O(n^3) where n is the number of elements of the input 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 Hliněný (2005) on checking monadic second-order formulas on 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.
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