Greedy algorithms and poset matroids

We generalize the matroid-theoretic approach to greedy algorithms to the setting of poset matroids, in the sense of Barnabei, Nicoletti and Pezzoli (1998) [BNP]. We illustrate our result by providing a generalization of Kruskal algorithm (which finds a minimum spanning subtree of a weighted graph) to abstract simplicial complexes

Matroids on convex geometries (cg-matroids)

AbstractWe consider matroidal structures on convex geometries, which we call cg-matroids. The concept of a cg-matroid is closely related to but different from that of a supermatroid introduced by Dunstan, Ingleton, and Welsh in 1972. Distributive supermatroids or poset matroids are supermatroids defined on distributive lattices or sets of order ideals of posets. The class of cg-matroids includes distributive supermatroids (or poset matroids). We also introduce the concept of a strict cg-matroid, which turns out to be exactly a cg-matroid that is also a supermatroid. We show characterizations of cg-matroids and strict cg-matroids by means of the exchange property for bases and the augmentation property for independent sets. We also examine submodularity structures of strict cg-matroids

COMs: Complexes of Oriented Matroids

In his seminal 1983 paper, Jim Lawrence introduced lopsided sets and featured them as asymmetric counterparts of oriented matroids, both sharing the key property of strong elimination. Moreover, symmetry of faces holds in both structures as well as in the so-called affine oriented matroids. These two fundamental properties (formulated for covectors) together lead to the natural notion of "conditional oriented matroid" (abbreviated COM). These novel structures can be characterized in terms of three cocircuits axioms, generalizing the familiar characterization for oriented matroids. We describe a binary composition scheme by which every COM can successively be erected as a certain complex of oriented matroids, in essentially the same way as a lopsided set can be glued together from its maximal hypercube faces. A realizable COM is represented by a hyperplane arrangement restricted to an open convex set. Among these are the examples formed by linear extensions of ordered sets, generalizing the oriented matroids corresponding to the permutohedra. Relaxing realizability to local realizability, we capture a wider class of combinatorial objects: we show that non-positively curved Coxeter zonotopal complexes give rise to locally realizable COMs.Comment: 40 pages, 6 figures, (improved exposition

Concepts in Action

This open access book is a timely contribution in presenting recent issues, approaches, and results that are not only central to the highly interdisciplinary field of concept research but also particularly important to newly emergent paradigms and challenges. The contributors present a unique, holistic picture for the understanding and use of concepts from a wide range of fields including cognitive science, linguistics, philosophy, psychology, artificial intelligence, and computer science. The chapters focus on three distinct points of view that lie at the core of concept research: representation, learning, and application. The contributions present a combination of theoretical, experimental, computational, and applied methods that appeal to students and researchers working in these fields

Concepts in Action

This open access book is a timely contribution in presenting recent issues, approaches, and results that are not only central to the highly interdisciplinary field of concept research but also particularly important to newly emergent paradigms and challenges. The contributors present a unique, holistic picture for the understanding and use of concepts from a wide range of fields including cognitive science, linguistics, philosophy, psychology, artificial intelligence, and computer science. The chapters focus on three distinct points of view that lie at the core of concept research: representation, learning, and application. The contributions present a combination of theoretical, experimental, computational, and applied methods that appeal to students and researchers working in these fields