Finding branch-decompositions of matroids, hypergraphs, and more

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

Ribbon graphs and bialgebra of Lagrangian subspaces

To each ribbon graph we assign a so-called L-space, which is a Lagrangian subspace in an even-dimensional vector space with the standard symplectic form. This invariant generalizes the notion of the intersection matrix of a chord diagram. Moreover, the actions of Morse perestroikas (or taking a partial dual) and Vassiliev moves on ribbon graphs are reinterpreted nicely in the language of L-spaces, becoming changes of bases in this vector space. Finally, we define a bialgebra structure on the span of L-spaces, which is analogous to the 4-bialgebra structure on chord diagrams.Comment: 21 pages, 13 figures. v2: major revision, Sec 2 and 3 completely rewritten; v3: minor corrections. Final version, to appear in Journal of Knot Theory and its Ramification

Characterization and Lower Bounds for Branching Program Size using Projective Dimension

We study projective dimension, a graph parameter (denoted by pd$(G)$ for a graph $G$), introduced by (Pudl\'ak, R\"odl 1992), who showed that proving lower bounds for pd$(G_f)$ for bipartite graphs $G_f$ associated with a Boolean function $f$ imply size lower bounds for branching programs computing $f$. Despite several attempts (Pudl\'ak, R\"odl 1992 ; Babai, R\'{o}nyai, Ganapathy 2000), proving super-linear lower bounds for projective dimension of explicit families of graphs has remained elusive. We show that there exist a Boolean function $f$ (on $n$ bits) for which the gap between the projective dimension and size of the optimal branching program computing $f$ (denoted by bpsize$(f)$), is $2^{\Omega(n)}$. Motivated by the argument in (Pudl\'ak, R\"odl 1992), we define two variants of projective dimension - projective dimension with intersection dimension 1 (denoted by upd$(G)$) and bitwise decomposable projective dimension (denoted by bitpdim$(G)$). As our main result, we show that there is an explicit family of graphs on $N = 2^n$ vertices such that the projective dimension is $O(\sqrt{n})$, the projective dimension with intersection dimension $1$ is $\Omega(n)$ and the bitwise decomposable projective dimension is $\Omega(\frac{n^{1.5}}{\log n})$. We also show that there exist a Boolean function $f$ (on $n$ bits) for which the gap between upd$(G_f)$ and bpsize$(f)$ is $2^{\Omega(n)}$. In contrast, we also show that the bitwise decomposable projective dimension characterizes size of the branching program up to a polynomial factor. That is, there exists a constant $c>0$ and for any function $f$, $\textrm{bitpdim}(G_f)/6 \le \textrm{bpsize}(f) \le (\textrm{bitpdim}(G_f))^c$. We also study two other variants of projective dimension and show that they are exactly equal to well-studied graph parameters - bipartite clique cover number and bipartite partition number respectively.Comment: 24 pages, 3 figure