20 research outputs found
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
Hierarchical decomposition of metabolic networks using k-modules
The optimal solutions obtained by flux balance analysis (FBA) are typically not unique. Flux modules have recently been shown to be a very useful tool to simplify and decompose the space of FBA-optimal solutions. Since yield-maximization is sometimes not the primary objective encountered in vivo, we are also interested in understanding the space of sub-optimal solutions. Unfortunately, the flux modules are too restrictive and not suited for this task. We present a generalization, called k-module, which compensates the limited applicability of flux modules to the space of sub-optimal solutions. Intuitively, a k-module is a sub-network with low connectivity to the rest of the network. Recursive application of k-modules yields a hierarchical decomposition of the metabolic network, which is also known as branch decomposition in matroid theory. In particular, decompositions computed by existing methods, like the null-space-based approach, introduced by Poolman et al. [(2007) J. Theor. Biol. 249, 691–705] can be interpreted as branch decompositions. With k-modules we can now compare alternative decompositions of metabolic networks to the classical sub-systems of glycolysis, tricarboxylic acid (TCA) cycle, etc. They can be used to speed up algorithmic problems [theoretically shown for elementary flux modes (EFM) enumeration] and have the potential to present computational solutions in a more intuitive way independently from the classical sub-systems
Computing with Tangles
Tangles of graphs have been introduced by Robertson and Seymour in the
context of their graph minor theory. Tangles may be viewed as describing
"k-connected components" of a graph (though in a twisted way). They play an
important role in graph minor theory. An interesting aspect of tangles is that
they cannot only be defined for graphs, but more generally for arbitrary
connectivity functions (that is, integer-valued submodular and symmetric set
functions).
However, tangles are difficult to deal with algorithmically. To start with,
it is unclear how to represent them, because they are families of separations
and as such may be exponentially large. Our first contribution is a data
structure for representing and accessing all tangles of a graph up to some
fixed order.
Using this data structure, we can prove an algorithmic version of a very
general structure theorem due to Carmesin, Diestel, Harman and Hundertmark (for
graphs) and Hundertmark (for arbitrary connectivity functions) that yields a
canonical tree decomposition whose parts correspond to the maximal tangles.
(This may be viewed as a generalisation of the decomposition of a graph into
its 3-connected components.
Partitions versus sets : a case of duality
In a recent paper, Amini et al. introduce a general framework to prove
duality theorems between special decompositions and their dual combinatorial
object. They thus unify all known ad-hoc proofs in one single theorem. While
this unification process is definitely good, their main theorem remains quite
technical and does not give a real insight of why some decompositions admit
dual objects and why others do not. The goal of this paper is both to
generalise a little this framework and to give an enlightening simple proof of
its central theorem
Matroid, Ideal, Ultrafilter, Tangle, and so on: Reconsideration of Obstruction to linear decomposition
The investigation of width parameters in both graph and algebraic contexts
has attracted considerable interest. Among these parameters, the linear branch
width has emerged as a crucial measure. In this concise paper, we explore the
concept of linear decomposition, specifically focusing on the single filter in
a connectivity system. Additionally, we examine the relevance of matroids,
antimatroids, and greedoids in the context of connectivity systems. Our primary
objective in this study is to shed light on the impediments to linear
decomposition from multiple perspectives.Comment: 11 page
Faster Algorithms For Vertex Partitioning Problems Parameterized by Clique-width
Many NP-hard problems, such as Dominating Set, are FPT parameterized by
clique-width. For graphs of clique-width given with a -expression,
Dominating Set can be solved in time. However, no FPT algorithm
is known for computing an optimal -expression. For a graph of clique-width
, if we rely on known algorithms to compute a -expression via
rank-width and then solving Dominating Set using the -expression,
the above algorithm will only give a runtime of . There
have been results which overcome this exponential jump; the best known
algorithm can solve Dominating Set in time by avoiding
constructing a -expression [Bui-Xuan, Telle, and Vatshelle. Fast dynamic
programming for locally checkable vertex subset and vertex partitioning
problems. Theoret. Comput. Sci., 2013. doi:10.1016/j.tcs.2013.01.009]. We
improve this to . Indeed, we show that for a graph of
clique-width , a large class of domination and partitioning problems
(LC-VSP), including Dominating Set, can be solved in . Our main tool is a variant of rank-width using the rank of a -
matrix over the rational field instead of the binary field.Comment: 13 pages, 5 figure