Tree automata and pigeonhole classes of matroids -- I

Hlineny's Theorem shows that any sentence in the monadic second-order logic of matroids can be tested in polynomial time, when the input is limited to a class of F-representable matroids with bounded branch-width (where F is a finite field). If each matroid in a class can be decomposed by a subcubic tree in such a way that only a bounded amount of information flows across displayed separations, then the class has bounded decomposition-width. We introduce the pigeonhole property for classes of matroids: if every subclass with bounded branch-width also has bounded decomposition-width, then the class is pigeonhole. An efficiently pigeonhole class has a stronger property, involving an efficiently-computable equivalence relation on subsets of the ground set. We show that Hlineny's Theorem extends to any efficiently pigeonhole class. In a sequel paper, we use these ideas to extend Hlineny's Theorem to the classes of fundamental transversal matroids, lattice path matroids, bicircular matroids, and H-gain-graphic matroids, where H is any finite group. We also give a characterisation of the families of hypergraphs that can be described via tree automata: a family is defined by a tree automaton if and only if it has bounded decomposition-width. Furthermore, we show that if a class of matroids has the pigeonhole property, and can be defined in monadic second-order logic, then any subclass with bounded branch-width has a decidable monadic second-order theory.Comment: Slightly extending the main theorem to cover a more expressive logi

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

The category of MSO transductions

MSO transductions are binary relations between structures which are defined using monadic second-order logic. MSO transductions form a category, since they are closed under composition. We show that many notions from language theory, such as recognizability or tree decompositions, can be defined in an abstract way that only refers to MSO transductions and their compositions

Tree automata and pigeonhole classes of matroids -- II

Let $\psi$ be a sentence in the counting monadic second-order logic of matroids. Let F be a finite field. Hlineny's Theorem says there is a fixed-parameter tractable algorithm for testing whether F-representable matroids satisfy $\psi$, with respect to the parameter of branch-width. In a previous paper we proved there is a similar fixed-parameter tractable algorithm for any efficiently pigeonhole class. In this sequel we apply results from the first paper and thereby extend Hlineny's Theorem to the classes of fundamental transversal matroids, lattice path matroids, bicircular matroids, and H-gain-graphic matroids, when H is a finite group. As a consequence, we can obtain a new proof of Courcelle's Theorem.Comment: Extending the main theorem slightly to cover a more expressive logi

Characterization of matrices with bounded graver bases and depth parameters and applications to integer programming

An intensive line of research on fixed parameter tractability of integer programming is focused on exploiting the relation between the sparsity of a constraint matrix A and the norm of the elements of its Graver basis. In particular, integer programming is fixed parameter tractable when parameterized by the primal tree-depth and the entry complexity of A, and when parameterized by the dual tree-depth and the entry complexity of A; both these parameterization imply that A is sparse, in particular, the number of its non-zero entries is linear in the number of columns or rows, respectively. We study preconditioners transforming a given matrix to an equivalent sparse matrix if it exists and provide structural results characterizing the existence of a sparse equivalent matrix in terms of the structural properties of the associated column matroid. In particular, our results imply that the \u1d4c1₁-norm of the Graver basis is bounded by a function of the maximum \u1d4c1₁-norm of a circuit of A. We use our results to design a parameterized algorithm that constructs a matrix equivalent to an input matrix A that has small primal/dual tree-depth and entry complexity if such an equivalent matrix exists. Our results yield parameterized algorithms for integer programming when parameterized by the \u1d4c1₁-norm of the Graver basis of the constraint matrix, when parameterized by the \u1d4c1₁-norm of the circuits of the constraint matrix, when parameterized by the smallest primal tree-depth and entry complexity of a matrix equivalent to the constraint matrix, and when parameterized by the smallest dual tree-depth and entry complexity of a matrix equivalent to the constraint matrix

First order convergence of matroids

The model theory based notion of the first order convergence unifies the notions of the left-convergence for dense structures and the Benjamini-Schramm convergence for sparse structures. It is known that every first order convergent sequence of graphs with bounded tree-depth can be represented by an analytic limit object called a limit modeling. We establish the matroid counterpart of this result: every first order convergent sequence of matroids with bounded branch-depth representable over a fixed finite field has a limit modeling, i.e., there exists an infinite matroid with the elements forming a probability space that has asymptotically the same first order properties. We show that neither of the bounded branch-depth assumption nor the representability assumption can be removed.Comment: Accepted to the European Journal of Combinatoric