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Hardness Magnification near State-Of-The-Art Lower Bounds Oliveira, Igor Carboni Pich, Ján Santhanam, Rahul Circuit Complexity Minimum Circuit Size Problem Kolmogorov Complexity This work continues the development of hardness magnification. The latter 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 <= s_1(N) from instances of complexity >= 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 super-polynomial lower bounds. Theorem. There exists a universal constant c >= 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}] !in Circuit[N^{1 + epsilon}], then NP !subseteq Circuit[poly]. (2) Gap-MKtP[2^{beta n}, 2^{beta n} + cn] !in TC^0[N^{1 + epsilon}], then EXP !subseteq TC^0[poly]. (3) Gap-MKtP[2^{beta n}, 2^{beta n} + cn] !in B_2-Formula[N^{2 + epsilon}], then EXP !subseteq Formula[poly]. (4) Gap-MKtP[2^{beta n}, 2^{beta n} + cn] !in U_2-Formula[N^{3 + epsilon}], then EXP !subseteq Formula[poly]. (5) Gap-MKtP[2^{beta n}, 2^{beta n} + cn] !in BP[N^{2 + epsilon}], then EXP !subseteq BP[poly]. (6) Gap-MKtP[2^{beta n}, 2^{beta n} + cn] !in (AC^0[6])[N^{1 + epsilon}], then EXP !subseteq AC^0[6]. 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 (3)-(5) 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 !subseteq NC^1 would follow via hardness magnification. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Igor Carboni Oliveira and Ján Pich and Rahul Santhanam 2019 Is Part Of LIPIcs, Volume 137, 34th Computational Complexity Conference (CCC 2019) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.CCC.2019.27 urn:nbn:de:0030-drops-108494 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.27 eng https://creativecommons.org/licenses/by/3.0/legalcode