Luca Ferrari
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.
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Let $\cX$ be a family of subsets of a finite set $E$. A matroid on $E$ is called an $\cX$-matroid if each set in $\cX$ is a circuit. We consider the problem of determining when there exists a unique maximal $\cX$-matroid in the weak order poset of all $\cX$-matroids on $E$, and characterizing its rank function when it exists.
Matroid schemes and geometric posets
本文通过定义几何偏序集和拟阵概形，刻画阿贝尔排列的交点组合结构，证明两者等价，并推广Tutte多项式至该框架。
Lattice of Integer Flows and the Poset of Strongly Connected Orientations for Regular Matroids
本文将Amini关于整数流格的Voronoi多面体面偏序集与子图强连通定向偏序集同构的结果推广至正则拟阵，并证明整数割定理可作为整数流定理在拟阵对偶性下的直接推论。
Dual Chow polynomials of matroids and posets
本文引入并研究弱秩偏序集关联代数中核的对偶Chow函数，重点分析拟阵特征函数的对偶Chow函数，证明了链公式、单峰性、gamma-正性以及拟阵的删除公式等结果。
We describe the structure of the monoid of natural-valued monotone functions on an arbitrary poset. For this monoid we provide a presentation, a characterization of prime elements, and a description of its convex hull. We also study the associated monoid ring, proving that it is normal, and thus Cohen-Macaulay. We determine its Cohen-Macaulay type, characterize the Gorenstein property, and provide a Gröbner basis of the defining ideal. Then we apply these results to the monoid of quasi-arithmetic multiplicities on a uniform matroid. Finally we state some conjectures on the number of irreducibl…
Modular flats of oriented matroids and poset quasi-fibrations
本文研究有向拟阵模平坦的组合学，证明到余秩一模平坦的局部化Salvetti复形的自然映射是偏序集拟纤维化，从而将超可解超平面排列的K(π,1)结果推广到超可解有向拟阵，并构造了许多不可实现的超可解有向拟阵。
Topological posets and tropical phased matroids
本文针对一大类拓扑偏序集证明了 McCord 定理的类比，即有序复形到 Up 拓扑偏序集的自然映射是弱同伦等价，并将其应用于热带相位拟阵，以阐明 MacPhersonian 猜想的热带版本。
In this paper we study general torus actions on manifolds with isolated fixed points from combinatorial point of view. The main object of study is the poset of face submanifolds of such actions. We introduce the notion of a locally geometric poset -- the graded poset locally modelled by geometric lattices, and prove that for any torus action, the poset of its faces is locally geometric. Next we discuss the relations between posets of faces and GKM-theory. In particular, we define the face poset of an abstract GKM-graph and show how to reconstruct the face poset of a manifold from its GKM-graph…
This is both an expository and research paper where we advocate a systematic study of continuous analogues of finite partially ordered sets, convex polytopes, oriented matroids, arrangements of subspaces, finite simplicial complexes, and other combinatorial structures. Among the illustrative examples are an Euler formula for a class of `continuous convex polytopes' (conjectured by Kalai and Wigderson), a duality result for a class of `continuous matroids', a calculation of the Euler characteristic of ideals in the Grassmannian poset (related to a problem of Gian-Carlo Rota), an exposition of t…
We study the representability problem for torsion-free arithmetic matroids. By using a new operation called "reduction" and a "signed Hermite normal form", we provide and implement an algorithm to compute all the representations, up to equivalence. As an application, we disprove two conjectures about the poset of layers and the independence poset of a toric arrangement.
Minimal matroids in dependency posets: algorithms and applications to computing irreducible decompositions of circuit varieties
本文研究点线构型及其极小拟阵与环路簇，提出一种在依赖序下识别极小拟阵的算法，用于计算环路簇的不可约分解，并应用于 Fano 等经典构型及探讨极小拟阵唯一性猜想。
Matroid bundles, introduced by MacPherson, are combinatorial analogues of real vector bundles. This paper sets up the foundations of matroid bundles, and defines a natural transformation from isomorphism classes of real vector bundles to isomorphism classes of matroid bundles, as well as a transformation from matroid bundles to spherical quasifibrations. The poset of oriented matroids of a fixed rank classifies matroid bundles, and the above transformations give a splitting from topology to combinatorics back to topology. This shows the mod 2 cohomology of the poset of rank k oriented matroids…
Chow polynomials of rank-uniform labeled posets
研究秩均匀标记偏序集的Chow多项式实根性质，揭示其与matroid代数几何结构的关联。
We introduce greedy weights of matroids, inspired by those for linear codes. We show that a Wei duality holds for two of these types of greedy weights for matroids. Moreover we show that in the cases where the matroids involved are associated to linear codes, our definitions coincide with those for codes. Thus our Wei duality is a generalization of that for linear codes given by Schaathun. In the last part of the paper we show how some important chains of cycles of the matroids appearing, correspond to chains of component maps of minimal resolutions of the independence complex of the correspon…
This letter studies the problem of minimizing increasing set functions, or equivalently, maximizing decreasing set functions, over the base of a matroid. This setting has received great interest, since it generalizes several applied problems including actuator and sensor placement problems in control theory, multi-robot task allocation problems, video summarization, and many others. We study two greedy heuristics, namely, the forward and the reverse greedy algorithms. We provide two novel performance guarantees for the approximate solutions obtained by these heuristics depending on both the su…
On the topology of conormal complexes and posets of matroids
本文研究了 matroid 的 biflats poset 的拓扑结构，证明了其与 conormal 复形和 Bergman 复形的 join 是简单同调等价的，并探讨了它们的组合性质。
Rook matroids and log-concavity of $P$-Eulerian polynomials
定义并研究车拟阵，其基对应斜费勒斯板上的非嵌套车放置，证明车拟阵是横截拟阵和正拟阵的子类，并利用非嵌套车放置的超对数凹性证明自然标号宽度2偏序集的$P$-欧拉多项式是超对数凹的。
Let the sign components of the maximal covectors of a simple oriented matroid M be represented by the real numbers -1 and 1. Consider the vertex set V(R) of a symmetric cycle R of adjacent topes in the tope graph of M as a subposet of the tope poset of M. If B is the bottom element of the tope poset then B is equal to the unweighted sum of the members of the set min V(R) of minimal elements of the subposet V(R); if B is the positive tope then the set min V(R) is a critical tope committee for the acyclic oriented matroid M.
We give an explicit description of the poset of cells of Bergman complexes of Lattice Path Matroids and establish a criterion for its simpliciality, in terms of the shape of the bounding paths.
The homological information about a filtered simplicial complex over the poset of positive real numbers is often presented by a barcode which depicts the evolution of the associated Betti numbers. However, there is a wonderfully complex combinatorics associated with the homology classes of a filtered complex, and one can do more than just counting them over the index poset. Here, we show that this combinatorial information can be encoded by filtered matroids, or even better, by rooted forests. We also show that these rooted forests can be realized as cobordisms.
The notion of $\mathcal{H}$-matroids was introduced by U. Faigle and S. Fujishige in 2009 as a general model for matroids and the greedy algorithm. They gave a characterization of $\mathcal{H}$-matroids by the greedy algorithm. In this note, we give a characterization of some $\mathcal{H}$-matroids by rank functions.
If V(R) is the vertex sequence of a symmetric cycle R in the tope graph of a simple acyclic oriented matroid M on a t-element ground set, then the set min V(R) of minimal elements in the subposet V(R) of the tope poset of M, based at the positive tope, is a critical committee for M that votes for the base tope. We consider the sequence zR of poset ranks of the elements from the vertex sequence of R as a fragment of a signal with period 2t and relate the number of members of the committee min V(R) to the magnitudes of [t/2] components, with odd indices, of the discrete Fourier transform of the …
In this work, we study matroids over a domain and several classical combinatorial and algebraic invariants related. We define their Grothendieck-Tutte polynomial $T_{\mathcal{M}}(x,y)$, extending the definition given by Fink and Moci in 2016, and we show that such polynomial has the classical deletion-contraction property. Moreover, we study the set of independencies for a realizable matroid over a domain, generalizing the definition of \emph{poset of torsions} $Gr(\mathcal{M})$ given by the second author in 2017. This is a union of identical simplicial posets as for (quasi-)arithmetic matroid…
Shellability of the quotient order on lattice path matroids
本文证明格路拟阵商序偏序集容许 EL-标号，从而证明其序复形可剥，并据此给出 Möbius 函数的界，且该标号恰为 Whitney 标号。
Sweeps, polytopes, oriented matroids, and allowable graphs of permutations
本文引入扫描定向拟阵与排列的容许图两种推广，以建模高维点构型的扫描偏序集，揭示了其与模超平面、Dilworth截断及广义Baues问题的联系，并证明伪扫描偏序集形变收缩为球面。
Coxeter matroids generalize matroids just as flag varieties of Lie groups generalize Grassmannians. Valuations of Coxeter matroids are functions that behave well with respect to subdivisions of a Coxeter matroid into smaller ones. We compute the universal valuative invariant of Coxeter matroids. A key ingredient is the family of Coxeter Schubert matroids, which correspond to the Bruhat cells of flag varieties. In the process, we compute the universal valuation of generalized Coxeter permutohedra, a larger family of polyhedra that model Coxeter analogues of combinatorial objects such as matroid…
Decomposition of Greedy Tamari Intervals and Bipartite Planar Maps
通过建立贪婪Tamari区间与二部平面地图的同构递归分解，给出了贪婪1-Tamari区间与平面2-星座等计数的组合证明，并提出了更一般的猜想。
We generalize the Varchenko matrix of a hyperplane arrangement to oriented matroids. We show that the celebrated determinant formula for the Varchenko matrix, first proved by Varchenko, generalizes to oriented matroids. It follows that the determinant only depends on the matroid underlying the oriented matroid and analogous formulas hold for cones in oriented matroids. We follow a proof strategy for the original Varchenko formula first suggested by Denham and Hanlon. Besides several technical lemmas this strategy also requires a topological result on supertopes which is of independent interest…
In recent years it has been noted that a number of combinatorial structures such as real and complex hyperplane arrangements, interval greedoids, matroids and oriented matroids have the structure of a finite monoid called a left regular band. Random walks on the monoid model a number of interesting Markov chains such as the Tsetlin library and riffle shuffle. The representation theory of left regular bands then comes into play and has had a major influence on both the combinatorics and the probability theory associated to such structures. In a recent paper, the authors established a close conn…
Generalized permutahedra are a family of polytopes with a rich combinatorial structure and strong connections to optimization. We prove that they are the universal family of polyhedra with a certain Hopf algebraic structure. Their antipode is remarkably simple: the antipode of a polytope is the alternating sum of its faces. Our construction provides a unifying framework to organize numerous combinatorial structures, including graphs, matroids, posets, set partitions, linear graphs, hypergraphs, simplicial complexes, building sets, and simple graphs. We highlight three applications: 1. We obtai…
Intervals in the greedy Tamari posets
本文研究 m-Dyck 路径上贪婪 Tamari 序的区间计数，通过递归方法和“催化”参数得到简单公式，并证明该公式同时计数 (m+1)-星座平面地图，还恢复了普通 m-Tamari 格的已知区间计数结果。
We introduce a greedy algorithm optimizing arrangements of lines with respect to a property. We apply this algorithm to the case of simpliciality: it recovers all known simplicial arrangements of lines in a very short time and also produces a yet unknown simplicial arrangement with 35 lines. We compute a (certainly incomplete) database of combinatorially simplicial complex arrangements of hyperplanes with up to 50 lines. Surprisingly, it contains several examples whose matroids have an infinite space of realizations up to projectivities.
The aim of the paper is to clarify the nature of combinatorial structures associated with maps on closed compact surfaces. We prove that maps give rise to Lagrangian matroids representable in a setting provided by cohomology of the surface with punctured points. Our proof is very elementary. We further observe that the greedy algorithm has a natural interpretation in this setting, as a `peeling' procedure which cuts the (connected) surface into a closed ring-shaped peel, and that this procedure is local.
Notes on the topology of independence structures
本文研究预独立空间中独立集偏序集及平坦集偏序集的拓扑性质，证明其在无限维时可缩、有限维时为 Cohen-Macaulay，从而推广了有限拟阵的独立性复形与几何格的相应结果。
Bernstein Transfers and Greedy Records for Fence and Circular-Fence Order Polynomials
研究关于围栏和圆形围栏序多项式，定义贪婪记录统计量，通过伯恩斯坦基证明相关等式，利用转移得到双射，构造循环记录，最终证明了卡哈内的圆形围栏猜想。
First-Fit is a greedy algorithm for partitioning the elements of a poset into chains. Let $\textrm{FF}(w,Q)$ be the maximum number of chains that First-Fit uses on a $Q$-free poset of width $w$. A result due to Bosek, Krawczyk, and Matecki states that $\textrm{FF}(w,Q)$ is finite when $Q$ has width at most $2$. We describe a family of posets $\mathcal{Q}$ and show that the following dichotomy holds: if $Q\in\mathcal{Q}$, then $\textrm{FF}(w,Q) \le 2^{c(\log w)^2}$ for some constant $c$ depending only on $Q$, and if $Q\not\in\mathcal{Q}$, then $\textrm{FF}(w,Q) \ge 2^w - 1$.
We present various new results on greedoids. We prove a theorem that generalizes an equivalent formulation of Edmonds' classic matroid polytope theorem to local forest greedoids -- a class of greedoids that contains matroids as well as branching greedoids. We also describe an application of this theorem in the field of measuring the reliability of networks by game-theoretical tools. Finally, we prove new results on the optimality of the greedy algorithm on greedoids and correct some mistakes that have been present in the literature for almost three decades.
We propose a novel approach for group elevator scheduling by formulating it as the maximization of submodular function under a matroid constraint. In particular, we propose to model the total waiting time of passengers using a quadratic Boolean function. The unary and pairwise terms in the function denote the waiting time for single and pairwise allocation of passengers to elevators, respectively. We show that this objective function is submodular. The matroid constraints ensure that every passenger is allocated to exactly one elevator. We use a greedy algorithm to maximize the submodular obje…
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