Matroid Research Articles

Liftable Point-Line Configurations: Defining Equations and Irreducibility of Associated Matroid and Circuit Varieties

We study point-line configurations through the lens of projective geometry and matroid theory. Our focus is on their realization spaces, where we introduce the concepts of liftable and quasi-liftable configurations, exploring cases in which an n-tuple of collinear points can be lifted to a nondegenerate realization of a point-line configuration. We show that forest configurations are liftable and characterize the realization space of liftable configurations as the solution set of certain linear systems of equations. Moreover, we study the Zariski closure of the realization spaces of liftable and quasi-liftable configurations, known as matroid varieties, and establish their irreducibility. Additionally, we compute an irreducible decomposition for their corresponding circuit varieties. Applying these liftability properties, we present a procedure to generate some of the defining equations of the associated matroid varieties. As corollaries, we provide a geometric representation for the defining equations of two specific examples: the quadrilateral set and the 3×4 grid. While the polynomials for the latter were previously computed using specialized algorithms tailored for this configuration, the geometric interpretation of these generators was missing. We compute a minimal generating set for the corresponding ideals.

Valuative invariants for large classes of matroids

AbstractWe study an operation in matroid theory that allows one to transition a given matroid into another with more bases via relaxing a stressed subset. This framework provides a new combinatorial characterization of the class of (elementary) split matroids. Moreover, it permits to describe an explicit matroid subdivision of a hypersimplex, which, in turn, can be used to write down concrete formulas for the evaluations of any valuative invariant on these matroids. This shows that evaluations on these matroids depend solely on the behavior of the invariant on a tractable subclass of Schubert matroids. We address systematically the consequences of our approach for several invariants. They include the volume and Ehrhart polynomial of base polytopes, the Tutte polynomial, Kazhdan–Lusztig polynomials, the Whitney numbers of the first and second kinds, spectrum polynomials and a generalization of these by Denham, chain polynomials and Speyer's ‐polynomials, as well as Chow rings of matroids and their Hilbert–Poincaré series. The flexibility of this setting allows us to give a unified explanation for several recent results regarding the listed invariants; furthermore, we emphasize it as a powerful computational tool to produce explicit data and concrete examples.

Efficient Representation of Lattice Path Matroids

Efficient Representation of Lattice Path Matroids

Computing Excluded Minors for Classes of Matroids Representable over Partial Fields

Computing Excluded Minors for Classes of Matroids Representable over Partial Fields

On Augmented Dimensional Analysis

We present an innovative approach to dimensional analysis, referred to as augmented dimensional analysis and based on a representation theorem for complete quantity functions with a scaling-covariant scalar representation. This new theorem, grounded in a purely algebraic theory of quantity spaces, allows the classical π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\pi $\\end{document} theorem to be restated in an explicit and precise form and its prerequisites to be clarified and relaxed. Augmented dimensional analysis, in contrast to classical dimensional analysis, is guaranteed to take into account all relations among the quantities involved. Several examples are given to show that the information thus gained, together with symmetry assumptions, can lead to new or stronger results. We also explore the connection between dimensional analysis and matroid theory, elucidating combinatorial aspects of dimensional analysis. It is emphasized that dimensional analysis rests on a principle of covariance.

Structural Synthesis of Planar Kinematic Chains for Non-Redundant Parallel Manipulators

Abstract This work focuses on the synthesis of non-redundant planar kinematic chains applied to parallel manipulators, where the mobility of the kinematic chains is constrained by the planar workspace, allowing for kinematic chains with up to three degrees-of-freedom. The synthesis process consists of two stages. First, a graph generator enumerates non-isomorphic biconnected graphs representing planar kinematic chains with up to three degrees-of-freedom and seven loops. Subsequently, graphs associated to chains with rigid subchains are removed using a degeneracy identification method based on matroid theory. Concise results are presented for chains with up to five loops. For chains with six and seven loops, results are categorized based on link partitions for a direct comparison with previous results. These findings align with prior research, with minor variations in the reported chain numbers. These variations can be attributed to several factors, including graph generation, isomorphism testing, fractionation, and subchain identification. These factors are comprehensively discussed and examined.

The Bhargava Greedoid as a Gaussian Elimination Greedoid

Inspired by Manjul Bhargava's theory of generalized factorials, Grinberg and Petrov have defined the Bhargava greedoid - a greedoid (a matroid-like set system on a finite set) assigned to any "ultra triple" (a somewhat extended variant of a finite ultrametric space). Here we show that the Bhargava greedoid of a finite ultra triple is always a Gaussian elimination greedoid over any sufficiently large (e.g., infinite) field; this is a greedoid analogue of a representable matroid. We find necessary and sufficient conditions on the size of the field to ensure this.

Line arrangements with many triple points

AbstractIn this paper, we construct an infinite series of line arrangements in characteristic two, each featuring only triple intersection points. This finding challenges the existing conjecture that suggests the existence of only a finite number of such arrangements, regardless of the characteristic. Leveraging the theory of matroids and employing computer algebra software, we rigorously examine the existence and non-existence across various characteristics of line arrangements with up to 19 lines maximizing the number of triple intersection points.

Labeled sample compression schemes for complexes of oriented matroids

We show that the topes of a complex of oriented matroids (abbreviated COM) of VC-dimension d admit a proper labeled sample compression scheme of size d. This considerably extends results of Moran and Warmuth on ample classes, of Ben-David and Litman on affine arrangements of hyperplanes, and of the authors on complexes of uniform oriented matroids, and is a step towards the sample compression conjecture – one of the oldest open problems in computational learning theory. On the one hand, our approach exploits the rich combinatorial cell structure of COMs via oriented matroid theory. On the other hand, viewing tope graphs of COMs as partial cubes creates a fruitful link to metric graph theory.

Matroid Lifts and Representability

A 1965 result of Crapo shows that every elementary lift of a matroid $M$ can be constructed from a linear class of circuits of $M$. In a recent paper, Walsh generalized this construction by defining a rank-$k$ lift of a matroid $M$ given a rank-$k$ matroid $N$ on the set of circuits of $M$, and conjectured that all matroid lifts can be obtained in this way. In this sequel paper we simplify Walsh's construction and show that this conjecture is true for representable matroids but is false in general. This gives a new way to certify that a particular matroid is non-representable, which we use to construct new classes of non-representable matroids.
 Walsh also applied the new matroid lift construction to gain graphs over the additive group of a non-prime finite field, generalizing a construction of Zaslavsky for these special groups. He conjectured that this construction is possible on three or more vertices only for the additive group of a non-prime finite field. We show that this conjecture holds for four or more vertices, but fails for exactly three.

On the Complexity of Chow and Hurwitz Forms

We consider the bit complexity of computing Chow forms of projective varieties defined over integers and their generalization to multiprojective spaces. We develop a deterministic algorithm using resultants and obtain a single exponential complexity upper bound. Earlier computational results for Chow forms were in the arithmetic complexity model; thus, our result represents the first bit complexity bound. We also extend our algorithm to Hurwitz forms in projective space and we explore connections between multiprojective Hurwitz forms and matroid theory. The motivation for our work comes from incidence geometry where intriguing computational algebra problems remain open.

Да цъфтят сто цветя, или за криптоморфизма

The paper discusses a possible way of justifying the existence of a multiplicity of cryptomorphic axiomatic theories. It is argued that what renders this multiplicity inevitable are the applications of the theory. They give rise to difficulties whose solution necessitates the application of different conceptual tools. The paper is organized as follows: (§1) introduces the concept of cryptomorphism and the canonic example for the phenomenon: matroid theory; (§2) discusses five cryptomorphic approaches to the definition of rationality in the framework of decision theory: through utility functions, weak orders, choice operators, layered permutations, and pop-stack sortability; (§3) shows that each of these different approaches can serve as a basis for the introduction of a different modification of the original theory. This establishes the heuristic value of cryptomorphic approaches in the domain of decision theory.

Representability of orthogonal matroids over partial fields

Representability of orthogonal matroids over partial fields

Chordal matroids arising from generalized parallel connections

A graph is chordal if every cycle of length at least four has a chord. In 1961, Dirac characterized chordal graphs as those graphs that can be built from complete graphs by repeated clique-sums. Generalizing this, we consider the class of simple GF(q)-representable matroids that can be built from projective geometries over GF(q) by repeated generalized parallel connections across projective geometries. We show that this class of matroids is closed under induced minors. We characterize the class by its forbidden induced minors; the case when q=2 is distinctive.

Sweeps, Polytopes, Oriented Matroids, and Allowable Graphs of Permutations

A sweep of a point configuration is any ordered partition induced by a linear functional. Posets of sweeps of planar point configurations were formalized and abstracted by Goodman and Pollack under the theory of allowable sequences of permutations. We introduce two generalizations that model posets of sweeps of higher dimensional configurations. Sweeps of a point configuration are in bijection with faces of an associated sweep polytope. Mimicking the fact that sweep polytopes are projections of permutahedra, we define sweep oriented matroids as strong maps of the braid oriented matroid. Allowable sequences are then the sweep oriented matroids of rank 2, and many of their properties extend to higher rank. We show strong ties between sweep oriented matroids and both modular hyperplanes and Dilworth truncations from (unoriented) matroid theory. Pseudo-sweeps are a generalization of sweeps in which the sweeping hyperplane is allowed to slightly change direction, and that can be extended to arbitrary oriented matroids in terms of cellular strings. We prove that for sweepable oriented matroids, sweep oriented matroids provide a sphere that is a deformation retract of the poset of pseudo-sweeps. This generalizes a property of sweep polytopes (which can be interpreted as monotone path polytopes of zonotopes), and solves a special case of the strong Generalized Baues Problem for cellular strings. A second generalization are allowable graphs of permutations: symmetric sets of permutations pairwise connected by allowable sequences. They have the structure of acycloids and include sweep oriented matroids.

Ehrhart theory of paving and panhandle matroids

Abstract We show that the base polytope P M of any paving matroid M can be systematically obtained from a hypersimplex by slicing off certain subpolytopes, namely base polytopes of lattice path matroids corresponding to panhandle-shaped Ferrers diagrams. We calculate the Ehrhart polynomials of these matroids and consequently write down the Ehrhart polynomial of P M , starting with Katzman’s formula for the Ehrhart polynomial of a hypersimplex. The method builds on and generalizes Ferroni’s work on sparse paving matroids. Combinatorially, our construction corresponds to constructing a uniform matroid from a paving matroid by iterating the operation of stressed-hyperplane relaxation introduced by Ferroni, Nasr and Vecchi, which generalizes the standard matroid-theoretic notion of circuit-hyperplane relaxation. We present evidence that panhandle matroids are Ehrhart positive and describe a conjectured combinatorial formula involving chain forests and Eulerian numbers from which Ehrhart positivity of panhandle matroids will follow. As an application of the main result, we calculate the Ehrhart polynomials of matroids associated with Steiner systems and finite projective planes, and show that they depend only on their design-theoretic parameters: for example, while projective planes of the same order need not have isomorphic matroids, their base polytopes must be Ehrhart equivalent.

Quantum algorithms for learning hidden strings with applications to matroid problems

In this paper, we explore quantum speedups for the problem, inspired by matroid theory, of learning a pair of n-bit binary strings that are promised to have the same number of 1s and differ in exactly two bits, by using the max inner product oracle and the sub-set oracle. More specifically, given two string s,s′∈{0,1}n satisfying the above constraints, for any x∈{0,1}n the max inner product oracle Omax(x) returns the max value between s⋅x and s′⋅x, and the sub-set oracle Osub(x) indicates whether the index set of the 1s in x is a subset of that in s or s′. We present an exact quantum algorithm consuming O(1) queries to the max inner product oracle for identifying the pair {s,s′}, and prove that any randomized algorithm requires Ω(n/log2⁡n) queries. Also, we present a quantum algorithm consuming n2+O(n) queries to the subset oracle, and prove that any randomized algorithm requires at least n−3+Ω(log2⁡n) queries. Therefore, quantum speedups are revealed in the two oracle models. Furthermore, the above results are applied to the problem in matroid theory of finding all the bases of a 2-bases matroid, where a matroid is called k-bases if it has k bases.

Mutations of Nucleic Acids via Matroidal Structures

The matroid concept is an important model in real life applications. Determining the existence of mutations of DNA and RNA plays an essential role in biological studies. The matroidal structures of matrices are used for determining the existence of mutations of DNA; graph theory and matroid theory can be used to identify important mutations in genetic data. We construct an algorithm to determine the existence of a mutation. Finally, we study the similarity and dissimilarity between genes using matroids.

Amalgamation of real zero polynomials

With this article, we hope to launch the investigation of what we call the Real Zero Amalgamation Problem. Whenever a polynomial arises from another polynomial by substituting zero for some of its variables, we call the second polynomial an extension of the first one. The Real Zero Amalgamation Problem asks when two (multivariate real) polynomials have a common extension (called amalgam) that is a real zero polynomial. We show that the obvious necessary conditions are not sufficient. Our counterexample is derived in several steps from a counterexample to amalgamation of matroids by Poljak and Turzík. On the positive side, we show that even a degree-preserving amalgamation is possible in three very special cases with three completely different techniques. Finally, we conjecture that amalgamation is always possible in the case of two shared variables. The analogue in matroid theory is true by another work of Poljak and Turzík. This would imply a very weak form of the Generalized Lax Conjecture.

Essence of independence: Hodge theory of matroids since June Huh

Matroids are combinatorial abstractions of independence, a ubiquitous notion that pervades many branches of mathematics. June Huh and his collaborators recently made spectacular breakthroughs by developing a Hodge theory of matroids that resolved several long-standing conjectures in matroid theory. We survey the main results in this development and ideas behind them.