We consider the question of whether worst-case hardness of the time-bounded Kolmogorov complexity problem, \(\textsf{MINK}^{\textsf{poly}}\) —that is, determining whether a string is time-bounded Kolmogorov random ( \(K^t\) -random) or not—suffices to imply the existence of one-way functions (OWF). Roughly speaking, our main result shows that under a natural strengthening of standard-type derandomization assumptions, worst-case hardness of the boundary version of this classic problem characterizes OWFs. In more detail, let \(\mathsf{boundary\text{- }MINK}^{t_1, t_2}\) denote the problem of, given an instance x, deciding whether (a) \(K^{t_2}(x)\ge n-1\) , or (b) \(K^{t_1}(x) < n-1\) but \(K^{t_2}> n - \log n\) ; that is, deciding whether x is \(K^t\) -random, or just “near” \(K^t\) -random. We say that \(\mathsf{boundary\text{- }MINK}^{\textsf{poly}}\notin \textsf{ioBPP}\) if \(\mathsf{boundary\text{- }MINK}^{\textsf{poly}}\notin \textsf{ioBPP}\) for all polynomials \(t_1,t_2\) . We show that under a natural strengthening of standard derandomization assumptions (namely, there exists a constant \(\varepsilon > 0\) such that \(\textsf{E}\not \subseteq \textsf{ioNTIME}[2^{kn}] /2^{\varepsilon n}\) for every \(k \in \mathbb {N}\) ), OWF exist iff \(\mathsf{boundary\text{- }MINK}^{\textsf{poly}}\notin \textsf{ioBPP}\) . Along the way, we also demonstrate that if we consider the probabilistic version of Kolmogorov complexity (referred to as \(pK^t\) ) instead, then the characterization holds unconditionally. We finally observe that for most standard optimization problems, hardness “along boundary” is equivalent to “plain” worst-case hardness, indicating that assuming hardness along the boundary may be WLOG.