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

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

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

Graph Decompositions and Monadic Second Order Logic

A tree decomposition is a tool which allows for analysis of the underlying tree structure of graphs which are not trees. Given a class of graphs with bounded tree width, many NP-complete problems can be computed in linear time for graphs in the class. Clique width of a graph G is a measure of the number of labels required to construct G using several particular graph operations. For any integer k, both the class of graphs with tree width at most k and the class of graphs with clique width at most k have a decidable monadic second order theory. In this paper we explore some recent results in applying these graph measures and their relation to monadic second order logic

Algorithmic Meta-Theorems

Algorithmic meta-theorems are general algorithmic results applying to a whole range of problems, rather than just to a single problem alone. They often have a "logical" and a "structural" component, that is they are results of the form: every computational problem that can be formalised in a given logic L can be solved efficiently on every class C of structures satisfying certain conditions. This paper gives a survey of algorithmic meta-theorems obtained in recent years and the methods used to prove them. As many meta-theorems use results from graph minor theory, we give a brief introduction to the theory developed by Robertson and Seymour for their proof of the graph minor theorem and state the main algorithmic consequences of this theory as far as they are needed in the theory of algorithmic meta-theorems

Parameterized complexity : permutation patterns, graph arrangements, and matroid parameters

The theory of parameterized complexity is an area of computer science focusing on refined analysis of hard algorithmic problems. In the thesis, we give two complexity lower bounds and define two novel parameters for matroids. The first lower bound is a kernelization lower bound for the Permutation Pattern Matching problem, which is concerned with finding a permutation pattern inside another input permutation. Our result states that unless a certain (widely believed) complexity hypothesis fails, it is impossible to construct a polynomial time algorithm taking an instance of the Permutation Pattern Matching problem and producing an equivalent instance of size bounded by a polynomial of the length of the pattern. Obtaining such lower bounds has been posed by Stephane Vialette as an open problem. We then prove a subexponential lower bound for the computational complexity of the Optimum Linear Arrangement problem. In our theorem, we assume a conjecture about the computational complexity of a variation of the Min Bisection problem. The two matroid parameters introduced in this work are called amalgam-width and branch-depth. Amalgam-width is a generalization of the branch-width parameter that allows for algorithmic applications even for matroids that are not finitely representable. We prove several results, including a theorem stating that deciding monadic second-order properties is fixed-parameter tractable for general matroids parameterized by amalgam-width. Branch-depth, the other newly introduced matroid parameter, is an analogue of graph tree-depth. We prove several statements relating graph tree-depth and matroid branch-depth. We also present an algorithm that efficiently approximates the value of the parameter on a general oracle-given matroid

The Linkage Problem for Group-labelled Graphs

This thesis aims to extend some of the results of the Graph Minors Project of Robertson and Seymour to "group-labelled graphs". Let $\Gamma$ be a group. A $\Gamma$-labelled graph is an oriented graph with its edges labelled from $\Gamma$, and is thus a generalization of a signed graph. Our primary result is a generalization of the main result from Graph Minors XIII. For any finite abelian group $\Gamma$, and any fixed $\Gamma$-labelled graph $H$, we present a polynomial-time algorithm that determines if an input $\Gamma$-labelled graph $G$ has an $H$-minor. The correctness of our algorithm relies on much of the machinery developed throughout the graph minors papers. We therefore hope it can serve as a reasonable introduction to the subject. Remarkably, Robertson and Seymour also prove that for any sequence $G_1, G_2, \dots$ of graphs, there exist indices $i<j$ such that $G_i$ is isomorphic to a minor of $G_j$. Geelen, Gerards and Whittle recently announced a proof of the analogous result for $\Gamma$-labelled graphs, for $\Gamma$ finite abelian. Together with the main result of this thesis, this implies that membership in any minor closed class of $\Gamma$-labelled graphs can be decided in polynomial-time. This also has some implications for well-quasi-ordering certain classes of matroids, which we discuss