Related papers: Coloring games and algebraic problems on matroids
With this work we aim to show how Mathematica can be a useful tool to investigate properties of combinatorial structures. Specifically, we will face enumeration problems on independent subsets of powers of paths and cycles, trying to…
Polymatroids are combinatorial abstractions of subspace arrangements in the same way that matroids are combinatorial abstractions of hyperplane arrangements. By introducing augmented Chow rings of polymatroids, modeled after augmented…
This is a survey on algorithmic questions about combinatorial and geometric properties of convex polytopes. We give a list of 35 problems; for each the current state of knowledege on its theoretical complexity status is reported. The…
En esta serie de tres articulos, damos una exposicion de varios resultados y problemas abiertos en tres areas de la combinatoria algebraica y geometrica: las matrices totalmente no negativas, las representaciones del grupo simetrico, y los…
Let k be a field of characteristic zero. We consider graded subalgebras A of k[x_1,...,x_m]/(x_1^2,...,x_m^2) generated by d linearly independant linear forms. Representations of matroids over k provide a natural description of the…
In this thesis I discuss combinatorial optimization problems, from the statistical physics perspective. The starting point are the motivations which brought physicists together with computer scientists and mathematicians to work on this…
We present a formal analysis, in Isabelle/HOL, of optimisation algorithms for matroids, which are useful generalisations of combinatorial structures that occur in optimisation, and greedoids, which are a generalisation of matroids. Although…
Combinatorial optimization can be described as the problem of finding a feasible subset that maximizes a objective function. The paper discusses combinatorial optimization problems, where for each dimension the set of feasible subsets is…
We show that the algebraic rank of divisors on certain graphs is related to the realizability problem of matroids. As a consequence, we produce a series of examples in which the algebraic rank depends on the ground field. We use the theory…
Existing approaches to solving combinatorial optimization problems on graphs suffer from the need to engineer each problem algorithmically, with practical problems recurring in many instances. The practical side of theoretical computer…
We survey recent results on combinatorial optimization problems in which the objective function is the entropy of a discrete distribution. These include the minimum entropy set cover, minimum entropy orientation, and minimum entropy…
We introduce combinatorial objects named matricubes that provide a generalization of the theory of matroids. As matroids provide a combinatorial axiomatization of hyperplane arrangements, matricubes provide a combinatorial axiomatization of…
We give a combinatorial characterization of generic frameworks that are minimally rigid under the additional constraint of maintaining symmetry with respect to a finite order rotation or a reflection. To establish these results we develop a…
Matroids and semigraphoids are discrete structures abstracting and generalizing linear independence among vectors and conditional independence among random variables, respectively. Despite the different nature of conditional independence…
We study compositions whose parts are colored by subsequences of the Fibonacci numbers. We give explicit bijections between Fibonacci colored compositions and several combinatorial objects, including certain restricted ternary and…
This article is a survey of matroid theory aimed at algebraic geometers. Matroids are combinatorial abstractions of linear subspaces and hyperplane arrangements. Not all matroids come from linear subspaces; those that do are said to be…
In this series of three articles, we give an exposition of various results and open problems in three areas of algebraic and geometric combinatorics: totally non-negative matrices, representations of the symmetric group, and hyperplane…
We define and study a collection of matroid isomorphism games corresponding to various axiomatic characterizations of matroids. These are nonlocal games played between two cooperative players. Each game is played on two matroids, and the…
In this paper we build an Orlik-Solomon model for the canonical gradation of the cohomology algebra with integer coefficients of the complement of a toric arrangement. We give some results on the uniqueness of the representation of…
We consider three forms of composition of matroids, each of which extends the category of bimatroids to a rigid monoidal category. Many well-known constructions are functorial or defined by morphisms in these categories. Motivating examples…