15 research outputs found
An Upper Bound on the Size of Obstructions for Bounded Linear Rank-Width
We provide a doubly exponential upper bound in on the size of forbidden
pivot-minors for symmetric or skew-symmetric matrices over a fixed finite field
of linear rank-width at most . As a corollary, we obtain a
doubly exponential upper bound in on the size of forbidden vertex-minors
for graphs of linear rank-width at most . This solves an open question
raised by Jeong, Kwon, and Oum [Excluded vertex-minors for graphs of linear
rank-width at most . European J. Combin., 41:242--257, 2014]. We also give a
doubly exponential upper bound in on the size of forbidden minors for
matroids representable over a fixed finite field of path-width at most .
Our basic tool is the pseudo-minor order used by Lagergren [Upper Bounds on
the Size of Obstructions and Interwines, Journal of Combinatorial Theory Series
B, 73:7--40, 1998] to bound the size of forbidden graph minors for bounded
path-width. To adapt this notion into linear rank-width, it is necessary to
well define partial pieces of graphs and merging operations that fit to
pivot-minors. Using the algebraic operations introduced by Courcelle and
Kant\'e, and then extended to (skew-)symmetric matrices by Kant\'e and Rao, we
define boundaried -labelled graphs and prove similar structure theorems for
pivot-minor and linear rank-width.Comment: 28 pages, 1 figur
Monadic transductions and definable classes of matroids
A transduction provides us with a way of using the monadic second-order
language of a structure to make statements about a derived structure. Any
transduction induces a relation on the set of these structures. This article
presents a self-contained presentation of the theory of transductions for the
monadic second-order language of matroids. This includes a proof of the matroid
version of the Backwards Translation Theorem, which lifts any formula applied
to the images of the transduction into a formula which we can apply to the
pre-images. Applications include proofs that the class of lattice-path matroids
and the class of spike-minors can be defined by sentences in monadic
second-order logic
Matrices of Optimal Tree-Depth and Row-Invariant Parameterized Algorithm for Integer Programming
A long line of research on fixed parameter tractability of integer programming culminated with showing that integer programs with n variables and a constraint matrix with tree-depth d and largest entry ? are solvable in time g(d,?) poly(n) for some function g, i.e., fixed parameter tractable when parameterized by tree-depth d and ?. However, the tree-depth of a constraint matrix depends on the positions of its non-zero entries and thus does not reflect its geometric structure. In particular, tree-depth of a constraint matrix is not preserved by row operations, i.e., a given integer program can be equivalent to another with a smaller dual tree-depth.
We prove that the branch-depth of the matroid defined by the columns of the constraint matrix is equal to the minimum tree-depth of a row-equivalent matrix. We also design a fixed parameter algorithm parameterized by an integer d and the entry complexity of an input matrix that either outputs a matrix with the smallest dual tree-depth that is row-equivalent to the input matrix or outputs that there is no matrix with dual tree-depth at most d that is row-equivalent to the input matrix. Finally, we use these results to obtain a fixed parameter algorithm for integer programming parameterized by the branch-depth of the input constraint matrix and the entry complexity. The parameterization by branch-depth cannot be replaced by the more permissive notion of branch-width
Monadic second-order model-checking on decomposable matroids
A notion of branch-width, which generalizes the one known for graphs, can be
defined for matroids. We first give a proof of the polynomial time
model-checking of monadic second-order formulas on representable matroids of
bounded branch-width, by reduction to monadic second-order formulas on trees.
This proof is much simpler than the one previously known. We also provide a
link between our logical approach and a grammar that allows to build matroids
of bounded branch-width. Finally, we introduce a new class of non-necessarily
representable matroids, described by a grammar and on which monadic
second-order formulas can be checked in linear time.Comment: 32 pages, journal paper. Revision: the last part has been removed and
the writing improve
Defining bicircular matroids in monadic logic
We conjecture that the class of frame matroids can be characterised by a
sentence in the monadic second-order logic of matroids, and we prove that there
is such a characterisation for the class of bicircular matroids. The proof does
not depend on an excluded-minor characterisation