119 research outputs found
From clutters to matroids
Peer ReviewedPostprint (published version
Linearly dependent vectorial decomposition of clutters
This paper deals with the question of completing a monotone increasing family of
subsets of a finite set
to obtain the linearly dependent subsets of a family of
vectors of a vector space. Specifically, we demonstrate that such vectorial completions
of the family of subsets ¿ exist and, in addition, we show that the minimal
vectorial completions of the family ¿ provide a decomposition of the clutter of the
inclusion-minimal elements of ¿. The computation of such vectorial decomposition
of clutters is also discussed in some cases.Peer ReviewedPostprint (author’s final draft
Transformation and decomposition of clutters into matroids
A clutter is a family of mutually incomparable sets. The set of circuits of a matroid, its set of bases, and its set of hyperplanes are examples of clutters arising from matroids. In this paper we address the question of determining which are the matroidal clutters that best approximate an arbitrary clutter ¿. For this, we first define two orders under which to compare clutters, which give a total of four possibilities for approximating ¿ (i.e., above or below with respect to each order); in fact, we actually consider the problem of approximating ¿ with clutters from any collection of clutters S, not necessarily arising from matroids. We show that, under some mild conditions, there is a finite non-empty set of clutters from S that are the closest to ¿ and, moreover, that ¿ is uniquely determined by them, in the sense that it can be recovered using a suitable clutter operation. We then particularize these results to the case where S is a collection of matroidal clutters and give algorithmic procedures to compute these clutters.Peer ReviewedPostprint (author's final draft
Powerful sets: a generalisation of binary matroids
A set of binary vectors, with positions indexed by ,
is said to be a \textit{powerful code} if, for all , the number
of vectors in that are zero in the positions indexed by is a power of
2. By treating binary vectors as characteristic vectors of subsets of , we
say that a set of subsets of is a \textit{powerful set} if
the set of characteristic vectors of sets in is a powerful code. Powerful
sets (codes) include cocircuit spaces of binary matroids (equivalently, linear
codes over ), but much more besides. Our motivation is that, to
each powerful set, there is an associated nonnegative-integer-valued rank
function (by a construction of Farr), although it does not in general satisfy
all the matroid rank axioms.
In this paper we investigate the combinatorial properties of powerful sets.
We prove fundamental results on special elements (loops, coloops, frames,
near-frames, and stars), their associated types of single-element extensions,
various ways of combining powerful sets to get new ones, and constructions of
nonlinear powerful sets. We show that every powerful set is determined by its
clutter of minimal nonzero members. Finally, we show that the number of
powerful sets is doubly exponential, and hence that almost all powerful sets
are nonlinear.Comment: 19 pages. This work was presented at the 40th Australasian Conference
on Combinatorial Mathematics and Combinatorial Computing (40ACCMCC),
University of Newcastle, Australia, Dec. 201
Completion and decomposition of a clutter into representable matroids
This paper deals with the question of completing a monotone increasing family of subsets Gamma of a finite set Omega to obtain the linearly dependent subsets of a family of vectors of a vector space. Specifically, we prove that such vectorial completions of the family of subsets Gamma exist and, in addition, we show that the minimal vectorial completions of the family Gamma provide a decomposition of the clutter Lambda of the inclusion-minimal elements of Gamma. The computation of such vectorial decomposition of clutters is also discussed in some cases. (C) 2015 Elsevier Inc. All rights reserved.Peer ReviewedPostprint (author’s final draft
Monotone clutters
A clutter is k-monotone, completely monotone or threshold if the corresponding Boolean function is k-monotone, completely monotone or threshold, respectively. A characterization of k-monotone clutters in terms of excluded minors is presented here. This result is used to derive a characterization of 2-monotone matroids and of 3-monotone matroids (which turn out to be all the threshold matroids). © 1993
Matroids are Immune to Braess Paradox
The famous Braess paradox describes the following phenomenon: It might happen
that the improvement of resources, like building a new street within a
congested network, may in fact lead to larger costs for the players in an
equilibrium. In this paper we consider general nonatomic congestion games and
give a characterization of the maximal combinatorial property of strategy
spaces for which Braess paradox does not occur. In a nutshell, bases of
matroids are exactly this maximal structure. We prove our characterization by
two novel sensitivity results for convex separable optimization problems over
polymatroid base polyhedra which may be of independent interest.Comment: 21 page
A unified interpretation of several combinatorial dualities
AbstractSeveral combinatorial structures exhibit a duality relation that yields interesting theorems, and, sometimes, useful explanations or interpretations of results that do not concern duality explicitly. We present a common characterization of the duality relations associated with matroids, clutters (Sperner families), oriented matroids, and weakly oriented matroids. The same conditions characterize the orthogonality relation on certain families of vector spaces. This leads to a notion of abstract duality
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