# Covering number

• The fractional covering number of (S,F) is $\min\{\sum_{Z \in F} x_Z:\ x \geq 0,\ \sum_{Z\in F: s \in Z} x_Z \geq 1\ \forall s \in S \}.$
• The circular covering number of (S,F) is the minimum circumference of a circle C for which it is possible to place the elements of S along the circle so that the elements placed in a unit interval $[x,x+1)$ are contained in a single member of F.
• Given $w: S \to {\mathbb Z}_+$, the $w$-covering number of (S,F) is $\min\{\sum_{Z \in F} x_Z:\ x \in {\mathbb Z}_+^F,\ \sum_{Z\in F: s \in Z} x_Z \geq w(s) \ \forall s \in S\}.$
In particular, a vertex cover of a graph G is a set of vertices that contains at leas one endpoint of every edge. Given $w: E \to {\mathbb Z}_+$, a $w$-vertex cover of G is a vector $x\in {\mathbb Z}_+^V$ such that $x_u+x_v \geq w(uv)$ for every $uv \in E$.