# How to construct an ordinary matching from a fractional matching?

Given a graph $$G=(V,E)$$. A fractional matching, say $$f$$, assigns every edge $$e \in E$$ to a fraction $$f(e) \in [0,1]$$, with the constraint: for $$v \in V$$, $$\sum_{e \ni v}f(e) \leq 1$$.

My question is : does the fractional matching give a matching that we know (of $$G$$) i.e. with no conflicting edges? I am confused, since there is no criteria for choosing a fraction $$f(e) \in [0,1]$$, for every $$e \in E$$. For example, if we have $$e_{1}, e_{2} \in E$$ such that $$v \in e_{1}, e_{2}$$ and I assign $$f(e_{1})=1/4$$ and $$f(e_{2})= 1/2$$, then $$f(e_{1})+f(e_{2})=3/4 \leq 1$$, hence both $$e_{1}$$ and $$e_{2}$$ should belong to the fractional matching.

It is not possible to get a maximum matching from any fractional matching by only looking at the values of the fractional matching. Consider the undirected triangle graph with vertices $$v_1,v_2,v_3$$ and edges $$e_1=(v_1,v_2), e_2=(v_1,v_3), e_3=(v_2,v_3)$$. If we assign $$f(e_1)=f(e_2)=f(e_3)=\frac{1}{2}$$, this is a valid fractional matching. However, an actual matching can only contain one of the edges, so we cannot distinguish these edges by their value.