17 research outputs found
Separations of Matroid Freeness Properties
Properties of Boolean functions on the hypercube invariant with respect to
linear transformations of the domain are among the most well-studied properties
in the context of property testing. In this paper, we study the fundamental
class of linear-invariant properties called matroid freeness properties. These
properties have been conjectured to essentially coincide with all testable
linear-invariant properties, and a recent sequence of works has established
testability for increasingly larger subclasses. One question left open,
however, is whether the infinitely many syntactically different properties
recently shown testable in fact correspond to new, semantically distinct ones.
This is a crucial issue since it has also been shown that there exist
subclasses of these properties for which an infinite set of syntactically
different representations collapse into one of a small, finite set of
properties, all previously known to be testable.
An important question is therefore to understand the semantics of matroid
freeness properties, and in particular when two syntactically different
properties are truly distinct. We shed light on this problem by developing a
method for determining the relation between two matroid freeness properties P
and Q. Furthermore, we show that there is a natural subclass of matroid
freeness properties such that for any two properties P and Q from this
subclass, a strong dichotomy must hold: either P is contained in Q or the two
properties are "well separated." As an application of this method, we exhibit
new, infinite hierarchies of testable matroid freeness properties such that at
each level of the hierarchy, there are functions that are far from all
functions lying in lower levels of the hierarchy. Our key technical tool is an
apparently new notion of maps between linear matroids, called matroid
homomorphisms, that might be of independent interest
Polytopal and structural aspects of matroids and related objects
PhDThis thesis consists of three self-contained but related parts. The rst is focussed on
polymatroids, these being a natural generalisation of matroids. The Tutte polynomial is
one of the most important and well-known graph polynomials, and also features prominently
in matroid theory. It is however not directly applicable to polymatroids. For
instance, deletion-contraction properties do not hold. We construct a polynomial for
polymatroids which behaves similarly to the Tutte polynomial of a matroid, and in fact
contains the same information as the Tutte polynomial when we restrict to matroids.
The second section is concerned with split matroids, a class of matroids which arises by
putting conditions on the system of split hyperplanes of the matroid base polytope. We
describe these conditions in terms of structural properties of the matroid, and use this
to give an excluded minor characterisation of the class.
In the nal section, we investigate the structure of clutters. A clutter consists of a nite
set and a collection of pairwise incomparable subsets. Clutters are natural generalisations
of matroids, and they have similar operations of deletion and contraction. We introduce
a notion of connectivity for clutters that generalises that of connectivity for matroids.
We prove a splitter theorem for connected clutters that has the splitter theorem for
connected matroids as a special case: if M and N are connected clutters, and N is a
proper minor of M, then there is an element in E(M) that can be deleted or contracted
to produce a connected clutter with N as a minor
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Proceedings of the 26th International Symposium on Theoretical Aspects of Computer Science (STACS'09)
The Symposium on Theoretical Aspects of Computer Science (STACS) is held alternately in France and in Germany. The conference of February 26-28, 2009, held in Freiburg, is the 26th in this series. Previous meetings took place in Paris (1984), Saarbr¨ucken (1985), Orsay (1986), Passau (1987), Bordeaux (1988), Paderborn (1989), Rouen (1990), Hamburg (1991), Cachan (1992), W¨urzburg (1993), Caen (1994), M¨unchen (1995), Grenoble (1996), L¨ubeck (1997), Paris (1998), Trier (1999), Lille (2000), Dresden (2001), Antibes (2002), Berlin (2003), Montpellier (2004), Stuttgart (2005), Marseille (2006), Aachen (2007), and Bordeaux (2008). ..
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
LIPIcs, Volume 248, ISAAC 2022, Complete Volume
LIPIcs, Volume 248, ISAAC 2022, Complete Volum
Conditional Gradient Methods
The purpose of this survey is to serve both as a gentle introduction and a
coherent overview of state-of-the-art Frank--Wolfe algorithms, also called
conditional gradient algorithms, for function minimization. These algorithms
are especially useful in convex optimization when linear optimization is
cheaper than projections.
The selection of the material has been guided by the principle of
highlighting crucial ideas as well as presenting new approaches that we believe
might become important in the future, with ample citations even of old works
imperative in the development of newer methods. Yet, our selection is sometimes
biased, and need not reflect consensus of the research community, and we have
certainly missed recent important contributions. After all the research area of
Frank--Wolfe is very active, making it a moving target. We apologize sincerely
in advance for any such distortions and we fully acknowledge: We stand on the
shoulder of giants.Comment: 238 pages with many figures. The FrankWolfe.jl Julia package
(https://github.com/ZIB-IOL/FrankWolfe.jl) providces state-of-the-art
implementations of many Frank--Wolfe method