Theory of Computing ------------------- Title : Hardness Magnification Near State-of-the-Art Lower Bounds Authors : Igor C. Oliveira, Jan Pich, and Rahul Santhanam Volume : 17 Number : 11 Pages : 1-38 URL : https://theoryofcomputing.org/articles/v017a011 Abstract -------- This article continues the development of hardness magnification, an emerging area that proposes a new strategy for showing strong complexity lower bounds by reducing them to a refined analysis of weaker models, where combinatorial techniques might be successful. We consider gap versions of the meta-computational problems MKtP and MCSP, where one needs to distinguish instances (strings or truth-tables) of complexity $\leq s_1(N)$ from instances of complexity $\geq s_2(N)$, and $N = 2^n$ denotes the input length. In MCSP, complexity is measured by circuit size, while in MKtP one considers Levin's notion of time-bounded Kolmogorov complexity. (In our results, the parameters $s_1(N)$ and $s_2(N)$ are asymptotically quite close, and the problems almost coincide with their standard formulations without a gap.) We establish that for Gap-MKtP[s_1,s_2] and Gap-MCSP[s_1,s_2], a _marginal improvement_ over the state of the art in unconditional lower bounds in a variety of computational models would imply explicit _superpolynomial_ lower bounds, including $P \neq NP$. Theorem. There exists a universal constant $c \geq 1$ for which the following hold. If there exists $\epsilon > 0$ such that for every small enough $\beta > 0$ * [(1)] Gap-MCSP$[2^{\beta n}/c n, 2^{\beta n}] \notin Circuit[N^{1 + \epsilon}]$, then $NP \nsubseteq Circuit[poly]$. * [(2)] Gap-MKtP$[2^{\beta n},\, 2^{\beta n} + cn] \notin B_2-Formula[N^{2 + \epsilon}]$, then $EXP \nsubseteq Formula[poly]$. * [(3)] Gap-MKtP$[2^{\beta n},\, 2^{\beta n} + cn] \notin U_2-Formula[N^{3 + \epsilon}]$, then $EXP \nsubseteq Formula[\mathsf{poly}]$. * [(4)] Gap-MKtP$[2^{\beta n},\, 2^{\beta n} + cn] \notin BP[N^{2 + \epsilon}]$, then $EXP \nsubseteq BP[\mathsf{poly}]$. These results are complemented by lower bounds for Gap-MCSP and Gap-MKtP against different models. For instance, the lower bound assumed in (1) holds for $U_2$-formulas of near-quadratic size, and lower bounds similar to (2)--(4) hold for various regimes of parameters. We also identify a natural computational model under which the hardness magnification threshold for Gap-MKtP lies _below_ existing lower bounds: $U_2$-formulas that can compute parity functions at the leaves (instead of just literals). As a consequence, if one managed to adapt the existing lower bound techniques against such formulas to work with Gap-MKtP, then $EXP \nsubseteq NC^1$ would follow via hardness magnification. -------------- A conference version of this paper appeared in the Proceedings of the 34th Computational Complexity Conference (CCC'19). A preprint of this article appeared in the Electronic Colloquium on Computational Complexity (ECCC) as Tech Report TR18-158.