Weighted matroid

Summary

In combinatorics, a branch of mathematics, a weighted matroid is a matroid endowed with a function that assigns a weight to each element. Formally, let $M = (E, I)$ be a matroid, where E is the set of elements and I is the family of independent set. A weighted matroid has a weight function $w : E \rightarrow \mathbb{R}^+$ for assigns a strictly positive weight to each element of $E$. We extend the function to subsets of $E$ by summation; $w(A)$ is the sum of $w(x)$ over $x$ in $A$.