One-way functions (OWFs) are central objects of study in cryptography and computational complexity theory. In a seminal work, Liu and Pass (FOCS 2020) proved that the average-case hardness of computing time-bounded Kolmogorov complexity is equivalent to the existence of OWFs. It remained an open problem to establish such an equivalence for the average-case hardness of some natural NP-complete problem. In this paper, we make progress on this question by studying a conditional variant of the Minimum KT-complexity Problem (MKTP), which we call McKTP, as follows. 1. First, we prove that if McKTP is average-case hard on a polynomial fraction of its instances, then there exist OWFs. 2. Then, we observe that McKTP is NP-complete under polynomial-time randomized reductions. 3. Finally, we prove that the existence of OWFs implies the nontrivial average-case hardness of McKTP. Thus the existence of OWFs is inextricably linked to the average-case hardness of this NP-complete problem. In fact, building on recent results of Ren and Santhanam (CCC 2021), we show that McKTP is hard-on-average if and only if there are logspace-computable OWFs.