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

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

Linearly dependent vectorial decomposition of clutters

This paper deals with the question of completing a monotone increas- ing family of subsets G of a finite set ¿ to obtain the linearly dependent subsets of a family of vectors of a vector space. Specifically, we demonstrate that such vec- torial completions of the family of subsets G exist and, in addition, we show that the minimal vectorial completions of the family G provide a decomposition of the clutter ¿ of the inclusion-minimal elements of G . The computation of such vectorial decomposition of clutters is also discussed in some casesPeer ReviewedPostprint (published version

From clutters to matroids

Peer ReviewedPostprint (published version

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

Cuboids, a class of clutters

The τ=2 Conjecture, the Replication Conjecture and the f-Flowing Conjecture, and the classification of binary matroids with the sums of circuits property are foundational to Clutter Theory and have far-reaching consequences in Combinatorial Optimization, Matroid Theory and Graph Theory. We prove that these conjectures and result can equivalently be formulated in terms of cuboids, which form a special class of clutters. Cuboids are used as means to (a) manifest the geometry behind primal integrality and dual integrality of set covering linear programs, and (b) reveal a geometric rift between these two properties, in turn explaining why primal integrality does not imply dual integrality for set covering linear programs. Along the way, we see that the geometry supports the τ=2 Conjecture. Studying the geometry also leads to over 700 new ideal minimally non-packing clutters over at most 14 elements, a surprising revelation as there was once thought to be only one such clutter

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