311 research outputs found
Chain minors are FPT
Given two finite posets P and Q, P is a chain minor of Q if there exists a
partial function f from the elements of Q to the elements of P such that for
every chain in P there is a chain C_Q in Q with the property that f restricted
to C_Q is an isomorphism of chains. We give an algorithm to decide whether a
poset P is a chain minor of o poset Q that runs in time O(|Q| log |Q|) for
every fixed poset P. This solves an open problem from the monograph by Downey
and Fellows [Parameterized Complexity, 1999] who asked whether the problem was
fixed parameter tractable
Successor-Invariant First-Order Logic on Graphs with Excluded Topological Subgraphs
We show that the model-checking problem for successor-invariant first-order
logic is fixed-parameter tractable on graphs with excluded topological
subgraphs when parameterised by both the size of the input formula and the size
of the exluded topological subgraph. Furthermore, we show that model-checking
for order-invariant first-order logic is tractable on coloured posets of
bounded width, parameterised by both the size of the input formula and the
width of the poset.
Our result for successor-invariant FO extends previous results for this logic
on planar graphs (Engelmann et al., LICS 2012) and graphs with excluded minors
(Eickmeyer et al., LICS 2013), further narrowing the gap between what is known
for FO and what is known for successor-invariant FO. The proof uses Grohe and
Marx's structure theorem for graphs with excluded topological subgraphs. For
order-invariant FO we show that Gajarsk\'y et al.'s recent result for FO
carries over to order-invariant FO
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
Are there any good digraph width measures?
Several different measures for digraph width have appeared in the last few
years. However, none of them shares all the "nice" properties of treewidth:
First, being \emph{algorithmically useful} i.e. admitting polynomial-time
algorithms for all \MS1-definable problems on digraphs of bounded width. And,
second, having nice \emph{structural properties} i.e. being monotone under
taking subdigraphs and some form of arc contractions. As for the former,
(undirected) \MS1 seems to be the least common denominator of all reasonably
expressive logical languages on digraphs that can speak about the edge/arc
relation on the vertex set.The latter property is a necessary condition for a
width measure to be characterizable by some version of the cops-and-robber game
characterizing the ordinary treewidth. Our main result is that \emph{any
reasonable} algorithmically useful and structurally nice digraph measure cannot
be substantially different from the treewidth of the underlying undirected
graph. Moreover, we introduce \emph{directed topological minors} and argue that
they are the weakest useful notion of minors for digraphs
Bernstein–Sato theory for determinantal ideals in positive characteristic
Sigui R un anell de polinomis sobre un cos k. Una varietat algebraica sobre k és un conjunt de punts on s'anul·len alguns polinomis de R. Amb la finalitat d'estudiar aquestes varietats i les seves singularitats, una pràctica comuna és construir invariants algebraics per a quantificar com de singular és la varietat. Aquest és precisament un dels objectius de la teoria de Bernstein-Sato. Durant els últims vint anys, la teoria en característica p > 0 ha vist un desenvolupament i creixement sense parangó. En aquest projecte calculem invariants de la teoria per ideals determinantals, és a dir, ideals generats pels determinants de submatrius de matrius genèriques d'indeterminades.Sea R un anillo de polinomios sobre un cuerpo k. Una variedad algebraica sobre k es un conjunto de puntos donde se anulan algunos polinomios de R. A fin de estudiar estas variedades y sus singularidades, una práctica común es construir invariantes algebraicos para cuantificar cómo de singular es la variedad. Este es precisamente uno de los objetivos de la teoría de Bernstein-Sato. Durante los últimos veinte años, la teoría en característica p > 0 ha visto un crecimiento y desarrollo sin parangón. En este proyecto calculamos invariantes de la teoría para ideales determinantales, a saber, ideales generados por los determinantes de submatrices de matrices genéricas de indeterminadas.Let R a polynomial ring over a field k. An algebraic variety over k is a set of points given as the zero loci of polynomials in R. In order to study these varieties and their singularities, a common practice is to construct algebraic invariants to quantify how singular the variety is. This is precisely one of the goals of Bernstein-Sato theory. In characteristic p > 0, the theory has seen unparalleled growth and development for the last twenty years. In this project we compute algebraic invariants of this theory for determinantal ideals, that is, ideals generated by the determinants of submatrices of generic matrices of indeterminates.Outgoin
On the pathwidth of almost semicomplete digraphs
We call a digraph {\em -semicomplete} if each vertex of the digraph has at
most non-neighbors, where a non-neighbor of a vertex is a vertex such that there is no edge between and in either direction.
This notion generalizes that of semicomplete digraphs which are
-semicomplete and tournaments which are semicomplete and have no
anti-parallel pairs of edges. Our results in this paper are as follows. (1) We
give an algorithm which, given an -semicomplete digraph on vertices
and a positive integer , in time either
constructs a path-decomposition of of width at most or concludes
correctly that the pathwidth of is larger than . (2) We show that there
is a function such that every -semicomplete digraph of pathwidth
at least has a semicomplete subgraph of pathwidth at least .
One consequence of these results is that the problem of deciding if a fixed
digraph is topologically contained in a given -semicomplete digraph
admits a polynomial-time algorithm for fixed .Comment: 33pages, a shorter version to appear in ESA 201
Graph Parameters, Universal Obstructions, and WQO
We introduce the notion of universal obstruction of a graph parameter, with
respect to some quasi-ordering relation. Universal obstructions may serve as
compact characterizations of the asymptotic behavior of graph parameters. We
provide order-theoretic conditions which imply that such a characterization is
finite and, when this is the case, we present some algorithmic implications on
the existence of fixed-parameter algorithms
A survey of parameterized algorithms and the complexity of edge modification
The survey is a comprehensive overview of the developing area of parameterized algorithms for graph modification problems. It describes state of the art in kernelization, subexponential algorithms, and parameterized complexity of graph modification. The main focus is on edge modification problems, where the task is to change some adjacencies in a graph to satisfy some required properties. To facilitate further research, we list many open problems in the area.publishedVersio
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