Related papers: Base Modulus for Matroid Truncation, Strength, and…
Homogeneous matroids are characterized by the property that strength equals fractional arboricity, and arise in the study of base modulus [22]. For graphic matroids, Cunningham [9] provided efficient algorithms for calculating graph…
In this work, we explore the application of modulus in matroid theory, specifically, the modulus of the family of bases of matroids. This study not only recovers various concepts in matroid theory, including the strength, fractional…
In this paper we consider the problem of finding the {\em densest} subset subject to {\em co-matroid constraints}. We are given a {\em monotone supermodular} set function $f$ defined over a universe $U$, and the density of a subset $S$ is…
Spanning tree modulus is a generalization of effective resistance that is closely related to graph strength and fractional arboricity. The optimal edge density associated with spanning tree modulus is known to produce two hierarchical…
For ${n \in \mathbb{N}}$, the $n$-truncation of a matroid $M$ of rank at least $n$ is the matroid whose bases are the $n$-element independent sets of $M$. One can extend this definition to negative integers by letting the $(-n)$-truncation…
Speyer recognized that matroids encode the same data as a special class of tropical linear spaces and Shaw interpreted tropically certain basic matroid constructions; additionally, Frenk developed the perspective of tropical linear spaces…
We introduce a new matroid (graph) invariant, the arboricity polynomial. Given a matroid, the arboricity polynomial enumerates the number of covers of the ground set by disjoint independent sets. We establish the polynomiality of the…
We give new characterizations for the class of uniformly dense matroids and study applications of these characterizations to graphic and real representable matroids. We show that a matroid is uniformly dense if and only if its base polytope…
The submodular partitioning problem asks to minimize, over all partitions $P$ of a ground set $V$, the sum of a given submodular function $f$ over the parts of $P$. The problem has seen considerable work in approximability, as it…
In this paper, we introduce the concept of Density-Balanced Subset in a matroid, in which independent sets can be sampled so as to guarantee that (i) each element has the same probability to be sampled, and (ii) those events are negatively…
Lorea [11] and later Frank et al. [8] generalized graphic matroids to hypergraphic matroids. In [8], the authors introduced hypertrees as a generalization of spanning trees and proved a form of the theorem of Tutte [18] and Nash-Williams…
We present an algebraic framework which simultaneously generalizes the notion of linear subspaces, matroids, valuated matroids, oriented matroids, and regular matroids. To do this, we first introduce algebraic objects called tracts which…
We study a family of matroid optimization problems with a linear constraint (MOL). In these problems, we seek a subset of elements which optimizes (i.e., maximizes or minimizes) a linear objective function subject to (i) a matroid…
In this paper, we consider dynamic matroids, where elements can be inserted to or deleted from the ground set over time. The independent sets change to reflect the current ground set. As matroids are central to the study of many…
Greedy minimum weight spanning tree packings have proven to be useful in connectivity-related problems. We study the process of greedy minimum weight base packings in general matroids and explore its applications. For general matroids, we…
We prove that the linear matroid that defines generic rigidity of $d$-dimensional body-rod-bar frameworks (i.e., structures consisting of disjoint bodies and rods mutually linked by bars) can be obtained from the union of ${d+1 \choose 2}$…
Many computational problems admit fast algorithms on special inputs, however, the required properties might be quite restrictive. E.g., many graph problems can be solved much faster on interval or cographs, or on graphs of small…
We investigate the complexity of counting trees, forests and bases of matroids from a parameterized point of view. It turns out that the problems of computing the number of trees and forests with $k$ edges are $\# W[1]$-hard when…
We derive the five-dimensional effective action of strongly coupled heterotic string theory for the complete (1,1) sector of the theory by performing a reduction, on a Calabi-Yau three-fold, of M-theory on S^1/Z_2. A crucial ingredient for…
Matroid intersection is a classical optimization problem where, given two matroids over the same ground set, the goal is to find the largest common independent set. In this paper, we show that there exists a certain "sparsifer": a subset of…