Hardness Magnification for Natural Problems

Oliveira, Igor Carboni; Santhanam, Rahul

doi:10.1109/focs.2018.00016

Cited by 33 publications

(46 citation statements)

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“…Most recently, the reference [19] proves Theorem 1.3 for deterministic formula lower bounds of size n 3+ε ; we generalize their results for probabilistic formula lower bounds of size n 2+ε . Our Theorem 1.5 generalizes n 2−ε -size formula lower bounds for MCSP [32,54,55] on larger circuit sizes. In [19,54,55] it is shown that an n 3+ε -size formula lower bound for MKtP[n o (1) ] implies EXP NC 1 , and [32,54] showed that MKtP[polylog(n)] does not have n 2−ε -size formulas.…”

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

confidence: 66%

“…Our Theorem 1.5 generalizes n 2−ε -size formula lower bounds for MCSP [32,54,55] on larger circuit sizes. In [19,54,55] it is shown that an n 3+ε -size formula lower bound for MKtP[n o (1) ] implies EXP NC 1 , and [32,54] showed that MKtP[polylog(n)] does not have n 2−ε -size formulas. That is, the "gap" between the magnification threshold and known lower bounds was n 3+ε versus n 2−ε .…”

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

confidence: 66%

“…Comparison With Prior Work. Several recent papers [19][20][21][53][54][55]] also establish hardness magnification results for problems such as MCSP and MKtP. Most recently, the reference [19] proves Theorem 1.3 for deterministic formula lower bounds of size n 3+ε ; we generalize their results for probabilistic formula lower bounds of size n 2+ε .…”

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

confidence: 81%

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

confidence: 66%

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

confidence: 66%

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

confidence: 81%

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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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“…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: 65%

The New Complexity Landscape Around Circuit Minimization

Allender

2020

Language and Automata Theory and Applications

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“…A recent line of research on hardness magnification[16,14] provides another motivation for proving relatively weak lower bounds for restricted circuit models against certain "gap variants" of MCSP. Such lower bounds are shown to imply much stronger (superpolynomial) lower bounds.…”

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confidence: 99%

Circuit Lower Bounds for MCSP from Local Pseudorandom Generators

Cheraghchi

Kabanets

et al. 2020

ACM Trans. Comput. Theory

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The Minimum Circuit Size Problem (MCSP) asks if a given truth table of a Boolean function f can be computed by a Boolean circuit of size at most θ, for a given parameter θ. We improve several circuit lower bounds for MCSP, using pseudorandom generators (PRGs) that are local; a PRG is called local if its output bit strings, when viewed as the truth table of a Boolean function, can be computed by a Boolean circuit of small size. We get new and improved lower bounds for MCSP that almost match the best-known lower bounds against several circuit models. Specifically, we show that computing MCSP, on functions with a truth table of length N , requires N 3−o(1)-size de Morgan formulas, improving the recent N 2−o(1) lower bound by Hirahara and Santhanam (CCC, 2017), N 2−o(1)-size formulas over an arbitrary basis or general branching programs (no non-trivial lower bound was known for MCSP against these models), and 2 Ω(N 1/(d+2.01))-size depth-d AC 0 circuits, improving the superpolynomial lower bound by Allender et al. (SICOMP, 2006). The AC 0 lower bound stated above matches the best-known AC 0 lower bound (for PARITY) up to a small additive constant in the depth. Also, for the special case of depth-2 circuits (i.e., CNFs or DNFs), we get an almost optimal lower bound of 2 N 1−o(1) for MCSP. 2012 ACM Subject Classification Theory of computation → Circuit complexity; Theory of computation → Pseudorandomness and derandomization Keywords and phrases minimum circuit size problem (MCSP), circuit lower bounds, pseudorandom generators (PRGs), local PRGs, de Morgan formulas, branching programs, constant depth circuits

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Hardness Magnification for Natural Problems

Abstract: Please refer to published version for the most recent bibliographic citation information. If a published version is known of, the repository item page linked to above, will contain details on accessing it.

Cited by 33 publications

References 50 publications

Sharp threshold results for computational complexity

Sharp threshold results for computational complexity

The New Complexity Landscape Around Circuit Minimization

Circuit Lower Bounds for MCSP from Local Pseudorandom Generators

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