Given a Cheeger inequality, $$2h_{G}\geq\lambda_{G}\geq\dfrac{h_{G}^{2}}{2}$$.
Cheeger inequality guarantees that the cut resulting from this efficient algorithm has the Cheeger ratio within a quadratic factor of the optimum.
My question is:
What is meant by 'within a quadratic factor of the optimum'?
I understand that $h_{G}\le2\sqrt{h_{G}}$. Thus, is the quadratic factor of optimal is influenced by the $\sqrt{h_{G}}$? And, does the constant '2' affect 'within a quadratic factor of optimality'. In another word, if the constant is replaced by another number such as '4', how does the optimality affected?
I am not a mathematician, thus I am not familiar with this concept. But as my work do involve some spectral clustering optimization, I need to understand this terminology.
Thanks in Advance.