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Shared independent set of two matroids
the matroid intersection problem is to find a largest common independent set in two matroids over the same ground set. If the elements of the matroid are
Matroid_intersection
Direct sum of uniform matroids
In mathematics, a partition matroid or partitional matroid is a matroid that is a direct sum of uniform matroids. It is defined over a base set in which
Partition_matroid
Abstraction of linear independence of vectors
In combinatorics, a matroid /ˈmeɪtrɔɪd/ is a structure that abstracts and generalizes the notion of linear independence in vector spaces. There are many
Matroid
Subdivision into few independent sets
and to compute the largest common independent set in the intersection of two given matroids. The arboricity of an undirected graph is the minimum number
Matroid_partitioning
Largest independent set of paired elements
generalization of graph matching and matroid intersection. It is also known as polymatroid matching, or the matchoid problem. Matroid parity can be solved in polynomial
Matroid_parity_problem
delta-matroid or Δ-matroid is a family of sets obeying an exchange axiom generalizing an axiom of matroids. A non-empty family of sets is a delta-matroid if
Delta-matroid
American/Canadian mathematician and computer scientist
proved the matroid intersection theorem, a very general combinatorial min-max theorem which, in modern terms, showed that the matroid intersection problem
Jack_Edmonds
Abstraction of ordered linear algebra
An oriented matroid is a mathematical structure that abstracts the properties of directed graphs, vector arrangements over ordered fields, and hyperplane
Oriented_matroid
Subroutine for testing independence
mathematics and computer science, a matroid oracle is a subroutine through which an algorithm may access a matroid, an abstract combinatorial structure
Matroid_oracle
Matroid with no linear representation
In mathematics, the Vámos matroid or Vámos cube is a matroid over a set of eight elements that cannot be represented as a matrix over any field. It is
Vámos_matroid
Partition of space by a hyperplanes
matroid (and has the same relationship to the intersection semilattice as does the matroid to the lattice in the lattice case), but is not a matroid if
Arrangement_of_hyperplanes
Abstraction of algebraic independence
and the flat generated by a set T of elements is the intersection of L with the field K[T]. A matroid that can be generated in this way is called algebraic
Algebraic_matroid
Matroid theory
Matroid-constrained number partitioning is a variant of the multiway number partitioning problem, in which the subsets in the partition should be independent
Matroid-constrained number partitioning
Matroid-constrained_number_partitioning
Join-meet algebra on matroid flats
In the mathematics of matroids and lattices, a geometric lattice is a finite atomistic semimodular lattice, and a matroid lattice is an atomistic semimodular
Geometric_lattice
American computer scientist (1933–1994)
2-coloring and 3-coloring for graphs, in the matroid intersection problem for intersections of two or three matroids, and in 2-SAT and 3-SAT for satisfiability
Eugene_Lawler
Maximal independent set of the matroid
In mathematics, a basis of a matroid is a maximal independent set of the matroid—that is, an independent set that is not contained in any other independent
Basis_of_a_matroid
constraints can be done in polynomial time, by reduction to the weighted matroid intersection problem. Gross-substitute utilities are more general than additive
Welfare_maximization
Problem in combinatorial optimization
problems that includes as special cases the minimum-cost flow problem, matroid intersection, and the problem of computing a minimum-weight dijoin in a weighted
Submodular_flow
Existence of a line through two points
oriented matroid with n {\displaystyle n} elements has at least 3 n / 7 {\displaystyle 3n/7} two-point lines, or equivalently every rank-3 matroid with fewer
Sylvester–Gallai_theorem
Set without nontrivial polynomial equalities
{\displaystyle T} of elements is the intersection of L {\displaystyle L} with the field K [ T ] {\displaystyle K[T]} . A matroid that can be generated in this
Algebraic_independence
Branch of discrete mathematics
Not only the structure but also enumerative properties belong to matroid theory. Matroid theory was introduced by Hassler Whitney and studied as a part
Combinatorics
On rearrangement of bases in matroids
In linear algebra and matroid theory, Rota's basis conjecture is an unproven conjecture concerning rearrangements of bases, named after Gian-Carlo Rota
Rota's_basis_conjecture
Realization of semialgebraic sets by points
algebraic (or semialgebraic) varieties as realization spaces of oriented matroids. Informally it can also be understood as the statement that point configurations
Mnëv's_universality_theorem
Partition of graph into sequence of paths
efficient graph algorithms. They may also be generalized from graphs to matroids. Several important classes of graphs may be characterized as the graphs
Ear_decomposition
Graph with at most one cycle per component
fact, they have at most as many edges as they have vertices) – and their matroid structure allows several other families of sparse graphs to be decomposed
Pseudoforest
Abstract simplicial complex describing a graph's cliques
every clique of the underlying graph Partition matroid, a kind of matroid whose matroid intersections may form clique complexes Bandelt & Chepoi (2008)
Clique_complex
General concept and operation in mathematics
matroid theory, the family of sets complementary to the independent sets of a given matroid themselves form another matroid, called the dual matroid.
Duality_(mathematics)
Mathematical ways to group elements of a set
geometric lattices and matroids, this lattice of partitions of a finite set corresponds to a matroid in which the base set of the matroid consists of the atoms
Partition_of_a_set
Geometry with 7 points and 7 lines
structure theory of matroids. Excluding the Fano plane as a matroid minor is necessary to characterize several important classes of matroids, such as regular
Fano_plane
In linear algebra, generated subspace
definition of the span of points in space, a subset X of the ground set of a matroid is called a spanning set if the rank of X equals the rank of the entire
Linear_span
- 3 Intersection of two partition matroids - 6.75 Intersection of a graphic matroid and a partition matroid - 10.66 General matroid with matroid rank
Bayesian-optimal_pricing
Set-to-real map with diminishing returns
vector. Matroid rank functions Let Ω = { e 1 , e 2 , … , e n } {\displaystyle \Omega =\{e_{1},e_{2},\dots ,e_{n}\}} be the ground set on which a matroid is
Submodular_set_function
Set that intersects every one of a family of sets
finite sets form the basis sets of a matroid, the transversal matroid of C. The independent sets of the transversal matroid are the partial transversals of
Transversal_(combinatorics)
Pseudolines arranged largely to study arrangements of lines
flip graph. Each rank-3 oriented matroid is equivalent to an arrangement of pseudolines, and each oriented matroid which is also uniform (in which the
Arrangement_of_pseudolines
Set system used in greedy optimization
a greedoid is a type of set system. It arises from the notion of the matroid, which was originally introduced by Whitney in 1935 to study planar graphs
Greedoid
Convex hull of a finite set of points in a Euclidean space
a bit-length which is not polynomial in this representation. Oriented matroid Nef polyhedron Steinitz's theorem for convex polyhedra Branko Grünbaum
Convex_polytope
Operation on the subsets of a set
of a relation is the smallest equivalence relation that contains it. In matroid theory, the closure of X is the largest superset of X that has the same
Closure_(mathematics)
Geometric structure of 8 points and 8 lines
as a matroid, whose elements are the points of the configuration and whose nontrivial flats are the lines of the configuration. In this matroid, a set
Möbius–Kantor_configuration
Mathematical system of orderings or sets
defining antimatroids as set systems are very similar to those of matroids, but whereas matroids are defined by an exchange axiom, antimatroids are defined instead
Antimatroid
Formulation of matroids using closure operators
and in full combinatorial pregeometry, are essentially synonyms for "matroid". They were introduced by Gian-Carlo Rota with the intention of providing
Pregeometry_(model_theory)
Algebraic encoding of graph connectivity
and number of connected components, with immediate generalizations to matroids. It is also the most general graph invariant that can be defined by a
Tutte_polynomial
Maximal subgraph whose vertices can reach each other
{\displaystyle n-c} is the matroid-theoretic rank of the graph, and the rank of its graphic matroid. The rank of the dual cographic matroid equals the circuit
Component_(graph_theory)
On forbidden minors in planar graphs
configurations) appear in a characterization of the graphic matroids by forbidden matroid minors. Wagner, K. (1937), "Über eine Eigenschaft der ebenen
Wagner's_theorem
Arrangement of hyperplanes
has a maximal flag consisting of modular elements. Equivalently, the intersection semilattice of the arrangement is a supersolvable lattice, in the sense
Supersolvable_arrangement
Skeletonized version of algebraic geometry
Ardila, Federico; Klivans, Caroline J. (2006). "The Bergman complex of a matroid and phylogenetic trees". Journal of Combinatorial Theory, Series B. 96
Tropical_geometry
Graph whose biconnected components are all cliques
in any graph may be found in polynomial time using an algorithm for the matroid parity problem. Since triangular cactus graphs are planar graphs, the largest
Block_graph
Set whose pairs have minima and maxima
algebras, Boolean algebras, distributive lattices, and geometric lattices (matroids). These lattice-like structures all admit order-theoretic as well as algebraic
Lattice_(order)
Graph divided into two independent sets
of bipartiteness to hypergraphs. Bipartite matroid, a class of matroids that includes the graphic matroids of bipartite graphs Bipartite network projection
Bipartite_graph
Mathematical object
(sets of size 2), and their vertices (sets of size 1). In the context of matroids and greedoids, abstract simplicial complexes are also called independence
Abstract_simplicial_complex
Describing a family of graphs by excluding certain (sub)graphs
finite obstruction set. Erdős–Hajnal conjecture Forbidden subgraph problem Matroid minor Zarankiewicz problem Diestel, Reinhard (2000), Graph Theory, Graduate
Forbidden graph characterization
Forbidden_graph_characterization
Affine subspace of a Euclidean space
example Dihedral angle (between two planes). See also Angles between flats.) Matroid Coplanarity Isometry Gallier, J. (2011). "Basics of Affine Geometry". Geometric
Flat_(geometry)
Result in combinatorics and graph theory
to determine the existence of a transversal which is independent in a matroid. Hall 1986, pg. 51. An alternative form of the marriage theorem applies
Hall's_marriage_theorem
Condition for 3 lines with common point to be perpendicular to the sides of triangle
Mathematisch für fortgeschrittene Anfänger : Weitere beliebte Beiträge von Matroids Matheplanet (in German). Heidelberg: Spektrum Akademischer Verlag. pp. 273–276
Carnot's theorem (perpendiculars)
Carnot's_theorem_(perpendiculars)
Any collection of sets, or subsets of a set
of a set in F {\displaystyle F} is also in F {\displaystyle F} . A matroid is an abstract simplicial complex with an additional property called the
Family_of_sets
Mathematical tree of cycles
in any graph may be found in polynomial time using an algorithm for the matroid parity problem. Since triangular cactus graphs are planar graphs, the largest
Cactus_graph
Trail in a graph that visits each edge once
non-empty intersection (the Eulerian graphs are both bridgeless and almost-Eulerian), but they do not contain each other. Eulerian matroid, an abstract
Eulerian_path
the graphic matroid of a graph, a subset of edges is independent if the corresponding subgraph is a tree or forest. In the bicircular matroid, a subset
Glossary_of_graph_theory
Embedding of a Grassmannian into projective space
Michel; Sturmfels, Bernd; White, Neil; Ziegler, Günter (1999), Oriented matroids, Encyclopedia of Mathematics and Its Applications, vol. 46 (2nd ed.), Cambridge
Plücker_embedding
of it include enumerative combinatorics, combinatorial design theory, matroid theory, extremal combinatorics and algebraic combinatorics, as well as
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Mathematics award
for geometric lattices, the proof of the Heron–Rota–Welsh conjecture for matroids, the development of the theory of Lorentzian polynomials, and the proof
Fields_Medal
Irrational system of points and lines
inherited by any other matroid within which the Perles matroid appears as a matroid minor. In tropical geometry, this matroid has been used to separate
Perles_configuration
Smallest convex set containing a given set
convex hulls may also be generalized in a more abstract way, to oriented matroids. It is not obvious that the first definition makes sense: why should there
Convex_hull
conjecture is still open. Matroid bundles. In 2003 Daniel Biss published a paper in the Annals of Mathematics claiming to show that matroid bundles are equivalent
List_of_incomplete_proofs
Method to solve optimization problems
Method of computing optimal strategies for last-success problems Oriented matroid – Abstraction of ordered linear algebra Quadratic programming – Solving
Linear_programming
Graded lattice with modular maximal chain
supersolvable, although it is not geometric. The lattice of flats of the graphic matroid for a graph is supersolvable if and only if the graph is chordal. Working
Supersolvable_lattice
the list of points in the intersection L ∩ V ( F ) {\displaystyle {\mathcal {L}}\cap {\mathbf {V} }(F)} . This intersection has finitely many points and
Numerical_algebraic_geometry
-graded sets. Signed sets are fundamental to the definition of oriented matroids. They may also be used to define the faces of a hypercube. If the hypercube
Signed_set
Mathematical ordering of a partial order
Günter M. (1992), "Introduction to Greedoids", in White, Neil (ed.), Matroid Applications, Encyclopedia of Mathematics and its Applications, vol. 40
Linear_extension
Mathematical operator
A and {x}. A finitary closure operator with this property is called a matroid. The dimension of a vector space, or the transcendence degree of a field
Closure_operator
Hungarian mathematician (born 1955)
independently published on the criss-cross algorithm. The theory of oriented matroids has also been used by Terlaky and Zhang (1991) to prove that their criss-cross
Tamás_Terlaky
Software for the algorithmic treatment of convex polyhedra
polyhedra, it is by now also capable of dealing with simplicial complexes, matroids, polyhedral fans, graphs, tropical objects, toric varieties and other objects
Polymake
Algorithmic problem of finding non-crossing drawings
planarity criterion that a graph is planar if and only if its graphic matroid is also cographic, Mac Lane's planarity criterion characterizing planar
Planarity_testing
Abstract strategy board game
mathematical underpinnings related to the Brouwer fixed-point theorem, matroids and graph connectivity. Hex is a finite, two-player perfect information
Hex_(board_game)
Economical computational problem
the bases of a matroid over the set of resources, then all best-response sequences converge in polynomial number of steps, and the matroid property is essential
Nash_equilibrium_computation
Indian-American mathematician
schemes defined by Kirchhoff polynomials to the representation spaces of matroids. Moreover, using Mnev's universality theorem, we show that these schemes
Prakash_Belkale
Concept from mathematical logic
an infinite matroid, or pregeometry. A model of a strongly minimal theory is determined up to isomorphism by its dimension as a matroid. Totally categorical
Strongly_minimal_theory
Analysis of datasets using techniques from topology
reduction can in fact be performed as the complex is constructed by using matroid theory, leading to further performance increases. Another recent algorithm
Topological_data_analysis
Sequence of spaces in linear algebra
. Filtration (mathematics) Flag (geometry) Flag manifold Grassmannian Matroid Kostrikin, Alexei I. and Manin, Yuri I. (1997). Linear Algebra and Geometry
Flag_(linear_algebra)
Branch of mathematics
those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises
Geometry
Dutch-Canadian engineer and designer
Federico Ardila, a Colombian mathematician specializing in combinatorics and matroid theory. In 2019, Khoe joined Ardila on a sabbatical which involved traveling
May-Li_Khoe
Branch of mathematics
Computational algebraic geometry is an area that has emerged at the intersection of algebraic geometry and computer algebra, with the rise of computers
Algebraic_geometry
minimums of finite collections of polynomials. Rota's basis conjecture: for matroids of rank n {\displaystyle n} with n {\displaystyle n} disjoint bases B i
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Game where groups of players may enforce cooperative behaviour
matroids. In this context, the core of a convex cost game is called the base polyhedron, because its elements generalize base properties of matroids.
Cooperative_game_theory
Geometric system of two mutually inscribed tetrahedra
the two configurations, including the fact that both are self-dual under Matroid duality. In abstract terms, the latter configuration has "points" 0, .
Möbius_configuration
Museum in Manhattan, New York
special traveling art installation dedicated to a celebration of the intersection of art and mathematics. The exhibit occupied a footprint approximately
National Museum of Mathematics
National_Museum_of_Mathematics
mathematician specializing in disease modeling Collette Coullard, American matroid theorist and operations researcher Judith Covington, American mathematics
List_of_women_in_mathematics
Field of knowledge
Coding theory, including error correcting codes and a part of cryptography Matroid theory Discrete geometry Discrete probability distributions Game theory
Mathematics
On when a space equals the closed convex hull of its extreme points
convex set containing S . {\displaystyle S.} It is also equal to the intersection of all closed convex subsets that contain S {\displaystyle S} and to
Krein–Milman_theorem
Branch of mathematics
Lasker-Noether theorem, namely that every ideal in a polynomial ring is a finite intersection of primary ideals. Macaulay proved the uniqueness of this decomposition
Abstract_algebra
Social choice problem
Munagala and Shah focus on three types of constraints: Matroid constraints: there is a fixed matroid M over the items, and the chosen items must form a basis
Multi-issue_voting
Quantified formulas with real-number variables
Michel; Sturmfels, Bernd; White, Neil; Ziegler, Günter M. (1993), Oriented Matroids, Encyclopedia of Mathematics and its Applications, vol. 46, Cambridge:
Existential theory of the reals
Existential_theory_of_the_reals
Concerned with the notion of stability in model theory
e. is prime and minimal over) a strongly minimal set, which carries a matroid structure determined by (model-theoretic) algebraic closure that gives
Stable_theory
Combinitorics of Polyhedra
facets are available. Abstract polytope Combinatorial commutative algebra Matroid polytope Order polytope Simplicial sphere Stable matching polytope Ziegler
Polyhedral_combinatorics
Counts tuples of non-intersecting lattice paths
2017-04-17 Lindström, Bernt (1973), On the vector representations of induced matroids, doi:10.1112/blms/5.1.85 Sagan, Bruce E. (2001), The symmetric group, Springer
Lindström–Gessel–Viennot lemma
Lindström–Gessel–Viennot_lemma
Branch of mathematics
analysis Measure theory Discrete Combinatorics Discrete geometry Graph theory Matroid theory Order theory Geometry Algebraic Affine Analytic Arithmetic Complex
Order_theory
Generalizations in graph theory
Thomas; Zhang, Yihao (2019-12-23), "A Tale of Santa Claus, Hypergraphs and Matroids", Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, Proceedings
Hall-type theorems for hypergraphs
Hall-type_theorems_for_hypergraphs
Algorithmically defined graph
group, an implicit model for group-theoretic algorithms Matroid oracle, an implicit model for matroid algorithms Korf, Richard E. (2008), "Linear-time disk-based
Implicit_graph
concepts in structural (combinatorial) rigidity theory, such as the rigidity matroid. The following results concern the l p p {\displaystyle l_{p}^{p}} -distance
Graph_flattenability
non-crossing spanning trees of planar point sets, and more generally bases of matroids, using a state space that swaps one edge for another. Euler tours in graphs
Reverse-search_algorithm
Type of graph in graph theory
"Exponentially many hypohamiltonian graphs", Graphs, Hypergraphs and Matroids III (Proc. Conf. Kalsk 1988), Zielona Góra: Higher College of Engineering
Hypohamiltonian_graph
MATROID INTERSECTION
MATROID INTERSECTION
Girl/Female
French Latin
From the Latin Lucretia. Famous bearer: Lucrece, a Roman matron who committed suicide as a public...
Girl/Female
Muslim
Leadress. Matron.
Girl/Female
Biblical
Rain, prison.
Boy/Male
Spanish
God's gift.
Boy/Male
Arabic, French, Hindu, Indian, Muslim
Rebellious; Ray of Light
Girl/Female
Arabic
Poetess; Matron
Girl/Female
Muslim
Poetess. Matron.
Girl/Female
Arabic, German, Hindu, Indian, Kannada, Marathi, Muslim, Telugu
Song; Poetess; Matron
Boy/Male
Biblical
Wand of government.
Female
Russian
(Матрона) Russian form of Latin Matrona, MATRYONA means "lady."
Boy/Male
Muslim
Rebellious
Boy/Male
Indian
Rebellious
Biblical
wand of government
Biblical
rain; prison
Girl/Female
Muslim
Leadress. Matron.
Girl/Female
Australian, British, Chinese, Christian, English, Jamaican, Portuguese
Compound of the Names Polly and Anna; Bitter; Gracious; One who Plays for Real Madrid
Girl/Female
Indian, Sanskrit
Mother
Female
Egyptian
, a wife of Rameses III.
MATROID INTERSECTION
MATROID INTERSECTION
Surname or Lastname
English
English : variant spelling of Kite.
Girl/Female
Hindu
Education
Girl/Female
Tamil
Girl/Female
Australian, French, German, Hebrew, Swedish
Pledged to God; My God is a Vow; God is My Oath
Girl/Female
German
Rule of Ice; Fair Lady
Surname or Lastname
English
English : variant spelling of Birks, itself a variant of Birch.
Surname or Lastname
English
English : unexplained.
Boy/Male
Hebrew Indian
Strong.
Boy/Male
Hindu
Honored, Chosen
Girl/Female
French, German, Greek, Irish, Latin
Light; Honor; Diminutive of Nora
MATROID INTERSECTION
MATROID INTERSECTION
MATROID INTERSECTION
MATROID INTERSECTION
MATROID INTERSECTION
n. pl.
The triangular, or maioid, crabs. See Illust. under Maioid, and Illust. of Spider crab, under Spider.
n. pl.
The maioid crabs.
n.
The point on the side of the skull where the lambdoid, parieto-mastoid and occipito-mastoid sutures.
a.
Same as Mastoid.
n.
The state of being a matron.
n.
The state of a matron.
n.
An old woman or matron.
a.
Same as Sauroid.
a.
Resembling a saurian superficially; as, a sauroid fish.
a.
Like the dartos; dartoic; as, dartoid tissue.
a.
Pertaining to, or in the region of, the mastoid process; mastoidal.
n.
A housekeeper; esp., a woman who manages the domestic economy of a public instution; a head nurse in a hospital; as, the matron of a school or hospital.
a.
Like a matron; sedate; grave; matronly.
n.
The collective body of matrons.
n.
See Matrix.
n.
See Matross.
n.
A native or inhabitant of Madrid.
n.
A mold; a matrix.
pl.
of Matrix