This question is inspired by a question posed by Shiva Kintali, Hardness of approximation assuming NP != coNP . Multiplication of two prime numbers of equal size is strong candidate for one-way function. We know the $P \ne NP$ does not imply the existence of one-way functions.
Are there any hardness of approximation results assuming the existence of one-way functions?
Ideally, assuming $P \neq NP$ would not be sufficient to prove such hardness of approximation result and we must assume that existence of one-way functions to prove such hardness of approximation result.