How do you prove a function is submodular?

A function f : 2N → R is said to be submodular, if it satisfies following property of diminishing marginal returns: for every A ⊆ B ⊆ N and j ∈ B, f(A ∪ {j}) − f(A) ≥ f(B ∪ {j}) − f(B). One way to understand submodularity is to think of f as a utility functions.

Is submodular function convex?

Theorem 2 (Lovász) A set function f : 2S → R with f(∅)=0 is submodular iff ˆf is convex.

What is a Matroid constraint?

Matroid Constraint: An independence family of particular interest is one induced by a matroid M = (N, I). Laminar matroids generalize partition matroids. We have a laminar family of sets on N and each set S in the family has an integer value kS. A set A ⊆ N is independent iff |A ∩ S| ≤ kS for each S in the family.

Is a set a function?

A set function is a function whose domain is a collection of sets. In many instances in real analysis, a set function is a function which associates an affinely extended real number to each set in a collection of sets.

What is a transversal matroid?

Transversal matroids are those matroids where the independent sets can be considered as the partial transversals of a family of sets. For a given transversal matroid such a family of sets is called a presentation of the matroid.

How do you prove something is a matroid?

A subset system is a matroid if it satisfies the exchange property: If i and i are sets in I and i has fewer elements than i , then there exists an element e ∈ i \ i such that i ∪ {e} ∈ I.

What do you mean by Matroids explain with suitable example?

To define a matroid from a graph, we’ll set the ground set E to be the set of edges. Then the independent sets will be those sets of edges that do not contain a cycle. For example, in the above graph, the red edges form an independent set, but the blue ones do not. The matroids you get this way are called graphic.

What is matroid in graph theory?

A matroid is a structure that generalizes the properties of indepen- dence. Relevant applications are found in graph theory and linear algebra. There are several ways to define a matroid, each relate to the concept of independence.

What are matroids used for?

In combinatorics, a branch of mathematics, a matroid /ˈmeɪtrɔɪd/ is a structure that abstracts and generalizes the notion of linear independence in vector spaces.