Finding Branch-Decompositions of Matroids, Hypergraphs, and More

Abstract

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 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 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 [J. Algorithms, 21 (1996), pp. 358-402] on tree-width of graphs. To extend their framework to branchdecompositions 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 Hlinex2c7;n acute accent y and Oum [SIAM J. Comput., 38 (2008), pp. 1012-1032] that runs in time O(n3) where n is the number of elements of the input 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 Hlinex2c7;n acute accent y [J. Combin. Theory Ser. B, 96 (2006), pp. 325-351] 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.