Polymatroid greedoids

AbstractThis paper discusses polymatroid greedoids, a superclass of them, called local poset greedoids, and their relations to other subclasses of greedoids. Polymatroid greedoids combine in a certain sense the different relaxation concepts of matroids as polymatroids and as greedoids. Some characterization results are given especially for local poset greedoids via excluded minors. General construction principles for intersection of matroids and polymatroid greedoids with shelling structures are given. Furthermore, relations among many subclasses of greedoids which are known so far, are demonstrated

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

Coloring Complexes and Combinatorial Hopf Monoids

We generalize the notion of coloring complex of a graph to linearized combinatorial Hopf monoids. These are a generalization of the notion of coloring complex of a graph. We determine when a combinatorial Hopf monoid has such a construction, and discover some inequalities that are satisfied by the quasisymmetric function invariants associated to the combinatorial Hopf monoid. We show that the collection of all such coloring complexes forms a combinatorial Hopf monoid, which is the terminal object in the category of combinatorial Hopf monoids with convex characters. We also study several examples of combinatorial Hopf monoids.Comment: 37 pages, 5 figure

A Note on Selectors and Greedoids

This note deals with relations between selectors studied by Henry Crapo and a special class of greedoids introduced by these authors in a previous paper. We show that selectors are greedoids with the interval property and that a second property of Crapo, which he calls ‘locally free’ is the interval property without upper bound. In the last section of the paper we show that retract sequences of posets are general greedoids, but not selectors

Pruning Processes and a New Characterization of Convex Geometries

We provide a new characterization of convex geometries via a multivariate version of an identity that was originally proved by Maneva, Mossel and Wainwright for certain combinatorial objects arising in the context of the k-SAT problem. We thus highlight the connection between various characterizations of convex geometries and a family of removal processes studied in the literature on random structures.Comment: 14 pages, 3 figures; the exposition has changed significantly from previous versio