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Questions tagged [matroids]

Matroids are a common generalization of linearly independent sets and independent sets in graphs. Among other applications, they are exactly the simplical complexes in which the greedy algorithm outputs the optimal solution. Matroids are also studied for their own sake.

A Matroid is a structure that abstracts and generalizes the notion of linear independence in vector spaces. There are many equivalent ways to define a matroid axiomatically, the most significant being in terms of: independent sets; bases or circuits; rank functions; closure operators; and closed sets or flats. In the language of partially ordered sets, a finite matroid is equivalent to a geometric lattice.

Matroid theory borrows extensively from the terminology of linear algebra and graph theory, largely because it is the abstraction of various notions of central importance in these fields. Matroids have found applications in geometry, topology, combinatorial optimization, network theory and coding theory

388 questions
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Smallest/Minimal bases of a topological space

The smallest possible cardinality of a base is called the weight of the topological space. I was wondering if all minimal bases have the same cardinality, and if every base contains a subset whose cardinality is the weight of the topological…
Tim
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How are inclusion-wise maximal and minimal sets defined?

I have tried to find them over the internet, but am lacking a resource that rigorously defines these two terms.
Aseem Dua
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Coloring a Generalized Latin Square

Suppose we have an $n \times n$ array, and there is a decomposition $\mathcal{A}$ of its coordinates $a_{i,j}$ into sets $A_m$ as follows: If $a_{i,j} \in A_m$, then $a_{j,i} \in A_m$. So they're symmetrical pieces of the array (across the…
John Samples
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Support of a vector

What is the support of a signed vector? By signed vector, I mean a vector which is determined by considering the signs of the coefficients of the entries of another vector.
Dan
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Applications of matroid theory.

I am considering learning about matroid theory. I would like to know what the applications of matroid theory are (if they exist) beforehand.
Asinomás
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Motivations for and applications of Matroid Theory?

I have taken an interest in this topic recently. If one is unfamiliar with matroids, I will give the definition here. Let $M=(E,\mathcal I)$ where $E$ is a finite set called the ground set and $\mathcal I$ is a collection of subsets of $E$ called…
Math1000
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Books about matroids

Could you recommend any approachable books/papers/texts about matroids (maybe a chapter from somewhere)? The ideal reference would contain multiple examples, present some intuitions and keep formalism to a necessary minimum. I would appreciate any…
dtldarek
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Pairs of matroids $(M_1, M_2)$ with $\mathcal C(M_1) = \mathcal B(M_2)$

Let $M$ be a matroid on (finite) ground set $E$ with $\mathcal B(M)$ as its set of bases and $\mathcal C(M)$ as its set of circuits. If we consider the uniform matroid $U_{m,n}$ for $m,n \in \mathbb N$ with $m < n$, then we see that $\mathcal…
Moritz
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What is the proof that the rank of a matroid is sub-modular?

Recall the definition of the rank of a matroid $(V, \mathcal{I})$: $$ r(A) = \operatorname{rank}(A) = \max_{I \in \mathcal{I}}\{ | A \cap I | \} = \max\{ |I| : I \subseteq A, I \in \mathcal{I} \}$$ I was trying to prove that the rank of a matroid is…
Charlie Parker
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Prescribing the dimension of intersections of sub-vector spaces

Let $n$ be a positive integer. For each subset $S$ of $\{1,\dots,n\}$ let $d_S$ be a nonnegative integer. Assume that the $(d_S)$ satisfy: $$ S\subset T\implies d_S\ge d_T, $$ $$ d_{S\cap T}\ge d_S+d_T-d_{S\cup T} $$ for all…
Pierre-Yves Gaillard
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Are there matroids with $\mathcal F(M) \subsetneq \mathcal F(M^*)$ or $\mathcal C(M) \subsetneq \mathcal C(M^*)$?

Let $M$ be a matroid with its dual matroid $M^*$. Moreover, $\mathcal C(M)$ denotes the set of circuits of $M$, and $\mathcal F(M)$ is the set of its flats. Question: Is there a matroid with $\mathcal F(M) \subsetneq \mathcal F(M^*)$ or $\mathcal…
Moritz
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Is there a connection between the "independent sets" in matroids and "independent sets" in graph theory?

I've been reading up on matroids recently, which are used in the theory of greedy algorithms. A matroid is a pair $(X, I)$ where $X$ is a set and $I \subseteq \wp(X)$ is a family of sets over $X$ called the independent sets in $X$. It occurred to me…
templatetypedef
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Equivalent definitions for a coloop?

From wikipedia, in a matroid, An element that belongs to no circuit is called a coloop. Equivalently, an element is a coloop if it belongs to every basis. I wonder why the equivalence? From the equivalence, I suspect that the union of all…
Tim
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Where can I find a proof of Tutte's theorem?

I dislike the proof of Kuratowski's theorem in my textbook, but the book mentions a theorem of William Tutte: Theorem: A graph $G$ is planar if and only if the conflict graph of each cycle of $G$ is bipartite. Where the conflict graph of the…
Jack M
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Assuming $G=(V(G),E(G))$ is a graph what does $\Delta(G)$ mean?

Perhaps someone is kind enough to explain to me the meaning of this mathematical symbol, that I found in Discrete Mathematics (Matroid Theory)? Let $G=(V(G),E(G))$ be a graph. What does $$\Delta(G)$$ mean? From the context I can determine, that…
Aufwind
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