Weak lower bounds on resource-bounded compression imply strong separations of complexity classes

McKay, Dylan M.; Murray, Cody D.; Williams, Ryan

doi:10.1145/3313276.3316396

Cited by 31 publications

(32 citation statements)

References 21 publications

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“…It is important to note, in this regard, that lower bounds have been proved for MCSP that essentially match the strongest lower bounds that we have for any problems in NP [21]. There is now a significant body of work, showing that slight improvements to those bounds, or other seemingly-attainable lower bounds for GapMKtP or GapMCSP or related problems, would yield dramatic complexity class separations [19,18,17,16,62,54,53,49].…”

Section: Magnificationsupporting

confidence: 52%

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Vaughan Jones, Kolmogorov Complexity, and the New Complexity Landscape around Circuit Minimization

Allender

2021

NZ J Math

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Section: Magnificationsupporting

confidence: 52%

“…• MCSP[2 n ]] requires time more than N 1.99 on any one-tape probabilistic Turing machine [20]. • If MCSP[2 δn ]] requires time more than N 1.01 on any one-tape deterministic Turing machine, then P = NP [49]. If it were not the case that δ < , this would yield a proof of P = NP.…”

Section: Magnificationmentioning

confidence: 99%

Vaughan Jones, Kolmogorov Complexity, and the New Complexity Landscape around Circuit Minimization

Allender

2021

NZ J Math

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“…Similar sharp threshold results were known before, but apparently only in cases where lower bound techniques are few. For example, [46] show MCSP[n o (1) ] SIZE[n 1+ε ] implies NP P /poly , and it is not hard to establish an Ω(n)-size lower bound for MCSP[n o (1) ]. However, in the case of general circuits, researchers have been stuck for decades on proving even a 6n-size lower bound for De Morgan circuits, 4 so an n 1+ε -size lower bound on MCSP seems far, relatively speaking.…”

Section: Theorem 15 (Sub-quadratic Mktp and Mcsp Lower Bounds)mentioning

confidence: 99%

Sharp threshold results for computational complexity

Chen

Jin

Williams

2020

Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing

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We establish several "sharp threshold" results for computational complexity. For certain tasks, we can prove a resource lower bound of n c for c ≥ 1 (or obtain an efficient circuit-analysis algorithm for n c size), there is strong intuition that a similar result can be proved for larger functions of n, yet we can also prove that replacing "n c " with "n c+ε " in our results, for any ε > 0, would imply a breakthrough n ω(1) lower bound. We first establish such a result for Hardness Magnification. We prove (among other results) that for some c, the Minimum Circuit Size Problem for (log n) c-size circuits on length-n truth tables (MCSP[(log n) c ]) does not have n 2−o(1)-size probabilistic formulas. We also prove that an n 2+ε lower bound for MCSP[(log n) c ] (for any ε > 0 and c ≥ 1) would imply major lower bound results, such as NP does not have n k-size formulas for all k, and #SAT does not have log-depth circuits. Similar results hold for time-bounded Kolmogorov complexity. Note that cubic size lower bounds are known for probabilistic De Morgan formulas (for other functions). Next we show a sharp threshold for Quantified Derandomization (QD) of probabilistic formulas: (a) For all α, ε > 0, there is a deterministic polynomial-time algorithm that finds satisfying assignments to every probabilistic formula of n 2−2α −ε size with at most 2 n α falsifying assignments. (b) If for some α, ε > 0, there is such an algorithm for probabilistic formulas of n 2−α +ε-size and 2 n α unsatisfying assignments, then a full derandomization of NC 1 follows: a deterministic poly-time algorithm additively approximating the acceptance probability of any polynomial-size formula. Consequently, NP does not have n k-size formulas, for all k. Finally we show a sharp threshold result for Explicit Obstructions, inspired by Mulmuley's notion of explicit obstructions from GCT. An explicit obstruction against S(n)-size formulas is a poly-time algorithm A such that A(1 n) outputs a list {(x i , f (x i))} i ∈[poly(n)] ⊆ {0, 1} n ×{0, 1}, and every S(n)-size formula F is inconsistent with the (partially defined) function f. We prove that for all ε > 0, there is an explicit obstruction against n 2−ε-size formulas, and prove that there is an explicit obstruction against n 2+ε-size formulas for some ε > 0 if and only if there is an explicit obstruction against all * L.C. and R.W. are supported by NSF CCF-1741615 and a Google Faculty Research Award. Portions of this work were completed while L.C. and R.W. were visiting the Simons Institute at UC Berkeley.

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“…It is important to note, in this regard, that lower bounds have been proved for MCSP that essentially match the strongest lower bounds that we have for any problems in NP [16]. There is now a significant body of work, showing that slight improvements to those bounds, or other seemingly-attainable lower bounds for GapMKtP or GapMCSP or related problems, would yield dramatic complexity class separations [12][13][14][15]34,38,39,45].…”

Section: Magnificationmentioning

confidence: 61%