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Matroid rank

Matroid rank is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Matroid rank rather than just read about it. In short: In the mathematical theory of matroids, the rank of a matroid is the maximum size of an independent set in the matroid. The rank of a subset S of elements of the matroid is, similarly, the maximum size of an independent subset of S, and the rank function of the matroid maps sets of elements to their ranks.

Matroid rank — main illustration
Matroid rank — illustration

Key takeaways

  • Matroid rank belongs to science; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Matroid rank to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Matroid rank from memory before moving on to harder problems.

Reference excerpt

In the mathematical theory of matroids, the rank of a matroid is the maximum size of an independent set in the matroid. The rank of a subset S of elements of the matroid is, similarly, the maximum size of an independent subset of S, and the rank function of the matroid maps sets of elements to their ranks. The rank function is one of the fundamental concepts of matroid theory via which matroids may be axiomatized. Matroid rank functions form an important subclass of the submodular set functions. The rank functions of matroids defined from certain other types of mathematical object such as undirected graphs, matrices, and field extensions are important within the study of those objects.

Examples In all examples, E is the base set of the matroid, and B is some subset of E.

Let M be the free matroid, where the independent sets are all subsets of E. Then the rank function of M is simply: r(B) = |B|. Let M be a uniform matroid, where the independent sets are the subsets of E with at most k elements, for some integer k. Then the rank function of M is: r(B) = min(k, |B|). Let M be a partition matroid: the elements of E are partitioned into categories, each category c has capacity kc, and the independent sets are those containing at most kc elements of category c. Then the rank function of M is: r(B) = sumc min(kc, |Bc|) where Bc is the subset B contained in category c. Let M be a graphic matroid, where the independent sets are all the acyclic edge-sets (forests) of some fixed undirected graph G. Then the rank function r(B) is the number of vertices in the graph, minus the number of connected components of B (including single-vertex components).

Properties and axiomatization The rank function of a matroid obeys the following properties. (R1) The value of the rank function is always a non-negative integer and the rank of the empty set is 0. (R2) For any two subsets A {\displaystyle A} and B {\displaystyle B} of E {\displaystyle E} , r ( A ∪ B ) + r ( A ∩ B ) ≤ r ( A ) + r ( B ) {\displaystyle r(A\cup B)+r(A\cap B)\leq r(A)+r(B)} . That is, the rank is a submodular set function. (R3) For any set A {\displaystyle A} and element x {\displaystyle x} , r ( A ) ≤ r ( A ∪ { x } ) ≤ r ( A ) + 1 {\displaystyle r(A)\leq r(A\cup \{x\})\leq r(A)+1} . These properties may be used as axioms to characterize the rank function of matroids: every integer-valued submodular set function on the subsets of a finite set that obeys the inequalities r ( A ) ≤ r ( A ∪ { x } ) ≤ r ( A ) + 1 {\displaystyle r(A)\leq r(A\cup \{x\})\leq r(A)+1} for all A {\displaystyle A} and x {\displaystyle x} is the rank function of a matroid. The above properties imply additional properties:

If A ⊂ B ⊂ E {\displaystyle A\subset B\subset E} , then r ( A ) ≤ r ( B ) ≤ r ( E ) {\displaystyle r(A)\leq r(B)\leq r(E)} . That is, the rank is a monotonic function.

r ( A ) ≤ | A | {\displaystyle r(A)\leq |A|} .

Other matroid properties from rank The rank function may be used to determine the other important properties of a matroid:

A set is independent if and only if its rank equals its cardinality, and dependent if and only if it has greater cardinality than rank. A nonempty set is a circuit if its cardinality equals one plus its rank and every subset formed by removing one element from the set has equal rank. A set is a basis if its rank equals both its cardinality and the rank of the matroid. A set is closed if it is maximal for its rank, in the sense that there does not exist another element that can be added to it while maintaining the same rank. The difference | A | − r ( A ) {\displaystyle |A|-r(A)} is called the nullity of the subset A {\displaystyle A} . It is the minimum number of elements that must be removed from A {\displaystyle A} to obtain an independent set. The corank of a subset A {\displaystyle A} can refer to at least two different quantities: some authors use it to refer to the rank of A {\displaystyle A} in the dual matroid, r ∗ ( A ) = | A | + r ( E ∖ A ) − r ( E ) {\displaystyle r^{*}(A)=|A|+r(E\setminus A)-r(E)} , while other authors use corank to refer to the difference r ( E ) − r ( A ) {\displaystyle r(E)-r(A)} .

… excerpt ends here. Continue reading the full article.

Illustrations

Matroid rank: The graphic matroid of a forest with 4 edges, which is the free matroid with a ground set of size 4 (also the uniform matroid U 4 4 {\displaystyle U{}_{4}^{4}} ). The rank of the given matroid is 4, the size of its largest independent set. As this matroid is a member of the graphic, free, and uniform matroid families, rank functions can be used to calculate the rank of the matroid, specifically r ( B ) = | B | {\displaystyle r(B)=|B|} as a free matroid, r ( B ) = min ( k , | B | ) {\displaystyle r(B)=\min(k,|B|)} as a uniform matroid, and the number of vertices in the graph, minus the number of connected components of B {\displaystyle B} (including single-vertex components), as a graphic matroid.
The graphic matroid of a forest with 4 edges, which is the free matroid with a ground set of size 4 (also the uniform matroid U 4 4 {\displaystyle U{}_{4}^{4}} ). The rank of the given matroid is 4, the size of its largest independent set. As this matroid is a member of the graphic, free, and uniform matroid families, rank functions can be used to calculate the rank of the matroid, specifically r ( B ) = | B | {\displaystyle r(B)=|B|} as a free matroid, r ( B ) = min ( k , | B | ) {\displaystyle r(B)=\min(k,|B|)} as a uniform matroid, and the number of vertices in the graph, minus the number of connected components of B {\displaystyle B} (including single-vertex components), as a graphic matroid.

Worked examples

Example 1 — a first encounter with Matroid rank

Start with the simplest possible case. Write down what Matroid rank claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Matroid rank before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Matroid rank ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Matroid rank

In research
Matroid rank appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Matroid rank in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Matroid rank is common in secondary-school and first-year university syllabi. It links to neighbouring topics Dimension, Matroid theory, so understanding it makes those chapters shorter.
In everyday life
Look for Matroid rank outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Matroid rank in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Matroid rank means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Matroid rank out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Matroid rank in simple terms?

In the mathematical theory of matroids, the rank of a matroid is the maximum size of an independent set in the matroid. The rank of a subset S of elements of the matroid is, similarly, the maximum size of an independent subset of S, and the rank function of the matroid maps sets of elements to thei…

Why does Matroid rank matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Matroid rank?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Matroid rank.

Tags

  • Dimension
  • Matroid theory

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