Graphic matroid
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Log-concave polynomials II: high-dimensional walks and an FPRAS for counting bases of a matroid Open
We design an FPRAS to count the number of bases of any matroid given by an independent set oracle, and to estimate the partition function of the random cluster model of any matroid in the regime where 0<q<1. Consequently, we can sample ran…
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Multi-Sensor Next-Best-View Planning as Matroid-Constrained Submodular Maximization Open
3D scene models are useful in robotics for tasks such as path planning, object manipulation, and structural inspection. We consider the problem of creating a 3D model using depth images captured by a team of multiple robots. Each robot sel…
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Matroids over hyperfields Open
We present an algebraic framework which simultaneously generalizes the notion of linear subspaces, matroids, valuated matroids, and oriented matroids. We call the resulting objects matroids over hyperfields. In fact, there are (at least) t…
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Log-concave polynomials, I: Entropy and a deterministic approximation algorithm for counting bases of matroids Open
We give a deterministic polynomial time $2^{O(r)}$-approximation algorithm for the number of bases of a given matroid of rank $r$ and the number of common bases of any two matroids of rank $r$. To the best of our knowledge, this is the fir…
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New Applications of m-Polar Fuzzy Matroids Open
Mathematical modelling is an important aspect in apprehending discrete and continuous physical systems. Multipolar uncertainty in data and information incorporates a significant role in various abstract and applied mathematical modelling a…
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Simplicial generation of Chow rings of matroids Open
We introduce a presentation of the Chow ring of a matroid by a new set of generators, called “simplicial generators.” These generators are analogous to nef divisors on projective toric varieties, and admit a combinatorial interpretation vi…
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From weakly separated collections to matroid subdivisions Open
We study arrangements of slightly skewed tropical hyperplanes, called blades by A. Ocneanu, on the vertices of a hypersimplex \(\Delta_{k,n}\), and we investigate the resulting induced polytopal subdivisions. We show that placing a blade o…
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Matroid And Its Relations Open
In this article, the connections amid matroid and other notions have been studied. The structure of matroid could be a reflection of some other structure in lattice theory, group theory, other algebraic structure, graph theory, combinatori…
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Regular Matroids Have Polynomial Extension Complexity Open
We prove that the extension complexity of the independence polytope of every regular matroid on n elements is O(n6). Past results of Wong and Martin on extended formulations of the spanning tree polytope of a graph imply a O(n2) bound for …
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Branch-depth: Generalizing tree-depth of graphs Open
We present a concept called the branch-depth of a connectivity function, that\ngeneralizes the tree-depth of graphs. Then we prove two theorems showing that\nthis concept aligns closely with the notions of tree-depth and shrub-depth of\ngr…
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Matroid And Its Outlines Open
In this article, there's an effort to make sense about the new versions of matroid. I believe that there's new idea on the background of. matroid. Two styles of matroid is defined in the background of fixed graphs and after that the attrib…
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Existence and Computation of Maximin Fair Allocations Under Matroid-Rank Valuations Open
We study fair and economically efficient allocation of indivisible goods among agents whose valuations are rank functions of matroids. Such valuations constitute a well-studied class of submodular functions (i.e., they exhibit a diminishin…
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Pure Exploration of Multi-armed Bandit Under Matroid Constraints Open
We study the pure exploration problem subject to a matroid constraint (Best-Basis) in a stochastic multi-armed bandit game. In a Best-Basis instance, we are given $n$ stochastic arms with unknown reward distributions, as well as a matroid …
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Combinatorial Secretary Problems with Ordinal Information Open
The secretary problem is a classic model for online decision making. Recently, combinatorial extensions such as matroid or matching secretary problems have become an important tool to study algorithmic problems in dynamic markets. Here the…
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Packing of arborescences with matroid constraints via matroid intersection Open
International audience
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Matroid relaxations and Kazhdan–Lusztig non-degeneracy Open
In this paper we study the interplay between the operation of circuit-hyperplane relaxation and the Kazhdan–Lusztig theory of matroids. We obtain a family of polynomials, not depending on the matroids but only on their ranks, that relate t…
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Dependencies Among Dependencies in Matroids Open
In 1971, Rota introduced the concept of derived matroids to investigate "dependencies among dependencies" in matroids. In this paper, we study the derived matroid $\delta M$ of an ${\mathbb F}$-representation of a matroid $M$. The matroid …
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Covering Intersecting Bi-set Families under Matroid Constraints Open
Edmonds's fundamental theorem on arborescences in [J. Edmonds, Edge-disjoint branchings, in Combinatorial Algorithms, Courant Comput. Sci. Sympos. 9, Algorithmics Press, New York, 1973, pp. 91--96] characterizes the existence of $k$ pairwi…
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Oriented matroids and combinatorial neural codes Open
A combinatorial neural code \({\mathscr C}\subseteq 2^{[n]}\) is called convex if it arises as the intersection pattern of convex open subsets of \(\mathbb{R}^d\). We relate the emerging theory of convex neural codes to the established the…
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Simplicial generation of Chow rings of matroids Open
We introduce a presentation of the Chow ring of a matroid by a new set of generators, called "simplicial generators." These generators are analogous to nef divisors on projective toric varieties, and admit a combinatorial interpretation vi…
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Common Information, Matroid Representation, and Secret Sharing for\n Matroid Ports Open
Linear information and rank inequalities as, for instance, Ingleton\ninequality, are useful tools in information theory and matroid theory. Even\nthough many such inequalities have been found, it seems that most of them\nremain undiscovere…
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Polyhedra and parameter spaces for matroids over valuation rings Open
In this paper we address two of the major foundational questions in the theory of matroids over rings. First, we provide a cryptomorphic axiomatisation, by introducing an analogue of the base polytope for matroids. Second, we describe a pa…
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Online Submodular Maximization with Free Disposal: Randomization Beats ¼ for Partition Matroids Open
We study the online submodular maximization problem with free disposal under a matroid constraint. Elements from some ground set arrive one by one in rounds, and the algorithm maintains a feasible set that is independent in the underlying …
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Basis shape loci and the positive Grassmannian Open
A basis shape locus takes as input data a zero/nonzero pattern in an $n \times k$ matrix, which is equivalent to a presentation of a transversal matroid. The locus is defined as the set of points in the Grassmannian of $k$ planes in $\math…
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Hypergraph characterization of split matroids Open
We provide a combinatorial study of split matroids, a class that was motivated by the study of matroid polytopes from a tropical geometry point of view. A nice feature of split matroids is that they generalize paving matroids, while being …
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Strict log-concavity of the Kirchhoff polynomial and its applications to the strong Lefschetz property Open
Anari, Gharan, and Vinzant proved (complete) log-concavity of the basis generating functions for all matroids. From the viewpoint of combinatorial Hodge theory, it is natural to ask whether these functions are "strictly" log-concave for si…
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Matroidal Entropy Functions: A Quartet of Theories of Information, Matroid, Design, and Coding Open
In this paper, we study the entropy functions on extreme rays of the polymatroidal region which contain a matroid, i.e., matroidal entropy functions. We introduce variable strength orthogonal arrays indexed by a connected matroid M and pos…
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Templates for Binary Matroids Open
A binary frame template is a device for creating binary matroids from graphic\nor cographic matroids. Such matroids are said to conform or coconform to the\ntemplate. We introduce a preorder on these templates and determine the\nnontrivial…
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Diverse Collections in Matroids and Graphs Open
We investigate the parameterized complexity of finding diverse sets of solutions to three fundamental combinatorial problems, two from the theory of matroids and the third from graph theory. The input to the Weighted Diverse Bases problem …
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The excluded minors for 2- and 3-regular matroids Open
The class of 2-regular matroids is a natural generalisation of regular and near-regular matroids. We prove an excluded-minor characterisation for the class of 2-regular matroids. The class of 3-regular matroids coincides with the class of …