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A theorem on random polynomials and some consequences in average complexity

1990
*
Journal of symbolic computation
*

*In*this paper we recall

*some*classical results

*on*the

*average*number of real roots (which is

*in*O(log n) where n is the degree of the

*polynomial*for many natural

*random*distributions)

*and*use them to get ... Real

*polynomials*have very often very few real roots,

*and*when algorithms depend

*on*the number of real roots of

*polynomials*rather than

*on*their degrees, this fact has

*consequences*

*on*

*average*

*complexity*...

*In*this paper we shall briefly recall

*some*results concerning the number of different real roots of

*a*

*random*

*polynomial*

*and*we shall use it to get estimates

*on*the

*average*

*complexity*of

*some*algorithms ...

##
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Some Results on Derandomization

2004
*
Theory of Computing Systems
*

If

doi:10.1007/s00224-004-1194-y
fatcat:67kyob6puzevhit4ov5dyztx2q
*A*is Σ p k -complete*and*NEXP*A*is*in*P*A*/poly then NEXP*A*= EXP = MA*A*. ... If NP is easy*on**average*then efficient pseudorandom generators exist*and*P = BPP. 2. ... Acknowledgments Thanks to Peter Bro Miltersen, Russell Impagliazzo, Dieter van Melkebeek, Valentine Kabanets*and*Osamu Watanabe for helpful discussions. ...##
###
Some Results on Derandomization
[chapter]

2003
*
Lecture Notes in Computer Science
*

If

doi:10.1007/3-540-36494-3_20
fatcat:g5suojh4fffrvimr5q4fixejle
*A*is Σ p k -complete*and*NEXP*A*is*in*P*A*/poly then NEXP*A*= EXP = MA*A*. ... If NP is easy*on**average*then efficient pseudorandom generators exist*and*P = BPP. 2. ... Acknowledgments Thanks to Peter Bro Miltersen, Russell Impagliazzo, Dieter van Melkebeek, Valentine Kabanets*and*Osamu Watanabe for helpful discussions. ...##
###
Average-case intractability vs. worst-case intractability

2004
*
Information and Computation
*

Furthermore, other

doi:10.1016/j.ic.2003.05.002
fatcat:v7cyr4ycljf4to6d4hh5haqt7e
*complexity*classes like P(PP)*and*PSPACE are shown to be intractable*on**average*unless they are easy*in*the worst case. ... We show that not all sets*in*NP (or other levels of the*polynomial*-time hierarchy) have efficient*average*-case algorithms unless the Arthur-Merlin classes MA*and*AM can be derandomized to NP*and*various ... Acknowledgments We are very grateful to Osamu Watanabe for permitting us to include part*one*of Corollary 7.2*in*the paper. ...##
###
Average-case intractability vs. worst-case intractability
[chapter]

1998
*
Lecture Notes in Computer Science
*

As

doi:10.1007/bfb0055799
fatcat:x4jqsloj4be2fmn5pdze6yglby
*a*further*consequence*we show for C 2 fP(PP); PSPACEg that C is not tractable*in*the*average*-case unless C = P. ... We use the assumption that all sets*in*NP (or other levels of the*polynomial*-time hierarchy) have e cient*average*-case algorithms to derive collapse*consequences*for MA, AM,*and*various subclasses of P ...*in*time*polynomial**on**average*(under the distribution induced by the*random*selfreduction) can be decided by*a**randomized*algorithm*in*expected*polynomial*time. ...##
###
Roots of the derivatives of some random polynomials

2011
*
Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation - SNC '11
*

Our observations show that the sets of real (respectively

doi:10.1145/2331684.2331703
dblp:conf/snc/Galligo11
fatcat:ar7zvksywvcmleo5ivwmacneu4
*complex*) roots of the derivatives of*some*classical families of*random**polynomials*admit*a*rich variety of patterns looking like discretized curves ... Then we present several conjectures*and*outline*a*strategy to explain the presented phenomena. ... Acknowledgments I am grateful to the anonymous reviewer for his helpful careful reading*and*his interesting suggestions. I thank Julien Barre*and*Didier Clamond for valuable remarks. ...##
###
Fine-Grained Derandomization: From Problem-Centric to Resource-Centric Complexity

2018
*
International Colloquium on Automata, Languages and Programming
*

problems)

doi:10.4230/lipics.icalp.2018.27
dblp:conf/icalp/CarmosinoIS18
fatcat:e45lnmtgxncpnbnaxkrafnmdou
*on**randomized*machines for all but finitely many input lengths, then we have the following derandomizations: BPP can be decided*in**polynomial*time using only n α*random*bits*on**average*over ... any efficient input distribution, for any constant α > 0 BPP can be decided*in**polynomial*time with no*randomness**on**average*over the uniform distribution This answers an open question of Ball et al. ... constant α,*and*an inverse*polynomial*function δ(N ) such that any*polynomial*-time*randomness*-reduced algorithm with coin bound N α fails*in*deciding L*on**average*over µ within δ(N ) error for infinitely ...##
###
Average-Case Complexity Versus Approximate Simulation of Commuting Quantum Computations

2016
*
Physical Review Letters
*

*One*conjecture relates to the hardness of estimating the

*complex*-temperature partition function for

*random*instances of the Ising model; the other concerns approximating the number of zeroes of

*random*... We observe that both conjectures can be shown to be valid

*in*the setting of worst-case

*complexity*. ... The precise values of the constants

*in*

*Theorem*1

*and*the above Conjectures are artifacts of the proof technique

*and*can be traded off against each other to

*some*extent:

*a*stronger

*average*-case hardness ...

##
###
Optimal Coding Theorems in Time-Bounded Kolmogorov Complexity
[article]

2022
*
arXiv
*
pre-print

running times of languages that are

arXiv:2204.08312v1
fatcat:g5y7win7wndgdguem6t5xxzqrq
*in**average**polynomial*-time over all P-samplable distributions. ... We consider pK^t*complexity*[GKLO22],*a*variant of rKt where the*randomness*is public*and*the time bound is fixed. ... We are grateful to Bruno Bauwens for discussions*and*useful insights. M. Zimand was supported*in*part by the National Science Foundation through grant CCF 1811729. Z. Lu*and*I.C. ...##
###
Pseudorandomness and the Minimum Circuit Size Problem

2020
*
Innovations in Theoretical Computer Science
*

We also show that for

doi:10.4230/lipics.itcs.2020.68
dblp:conf/innovations/Santhanam20
fatcat:pgatorc5k5hlrknytijhfgjufy
*a*certain natural variant of MCSP, there is*a**polynomial*-time reduction from approximating the problem well*in*the worst case to solving it*on**average*. ... circuits with membership queries over the uniform distribution is hard; MCSP[2 n ] is zero-error hard*on**average*for*some*> 0; Cryptographic succinct hitting set generators exist. 3. ... We show how to solve AveMCSP[2 n , 2 δn ]*in**polynomial*size, for*some*fixed during the argument which depends*on*δ*and*k. ...##
###
DERANDOMIZATION: A BRIEF OVERVIEW
[chapter]

2004
*
Current Trends in Theoretical Computer Science
*

This survey focuses

doi:10.1142/9789812562494_0012
fatcat:znfvduwgrjexpcbcj62ientmvy
*on*the recent (1998)(1999)(2000)(2001)(2002)(2003) developments*in*the area of derandomization, with the emphasis*on*the derandomization of time-bounded*randomized**complexity*classes ... Acknowledgments I want to thank Lance Fortnow, Oded Goldreich, Russell Impagliazzo, Dieter van Melkebeek, Chris Umans, Salil Vadhan,*and*Avi Wigderson for*a*number of helpful comments*and*suggestions that ... Miltersen*and*Vinodchandran [MV99] improve upon*Theorem*4 by replacing the assumption of high SAT-oracle circuit*complexity*with that of high nondeterministic circuit*complexity*; the*average*-case version ...##
###
Condition Numbers for the Cube. I: Univariate Polynomials and Hypersurfaces
[article]

2021
*
arXiv
*
pre-print

*On*the

*one*hand, we show that the

*average*run-time of the Plantinga-Vegter algorithm is

*polynomial*

*in*the degree for

*random*sparse (alas

*a*restricted sparseness structure)

*polynomials*

*and*

*random*Gaussian ... We initiate

*a*condition-based

*complexity*framework based

*on*the norm of the cube that is

*a*step

*in*this direction. We present this framework for real hypersurfaces

*and*univariate

*polynomials*. ... Acknowledgments The first author is supported by

*a*postdoctoral fellowship of the 2020 "Interaction" program of the Fondation Sciences Mathématiques de Paris. ...

##
###
Hardness Magnification for Natural Problems

2018
*
2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)
*

We show that if MCSP[2 √ n ] cannot be solved

doi:10.1109/focs.2018.00016
dblp:conf/focs/OliveiraS18
fatcat:4youvi7wjfbxzn32ttjuy4c6f4
*on**average*with zero error by formulas of linear (or even sub-linear) size, then NP does not have*polynomial*-size formulas. ...*In*contrast, Hirahara*and*Santhanam [1] recently showed that MCSP[2 ... ACKNOWLEDGEMENTS We are grateful to Jan Krajíček, Ján Pich*and*Ninad Rajgopal for helpful discussions*and*comments. ...##
###
Pseudodeterministic algorithms and the structure of probabilistic time

2021
*
Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing
*

*A*new unconditional pseudorandom generator

*and*its

*consequences*. ... We build

*on*techniques developed to prove hierarchy

*theorems*for probabilistic time with advice (Fortnow

*and*Santhanam [FS04]) to construct an unconditional pseudorandom generator computable

*in*pseudodeterministic ... But using

*Theorem*12

*and*proceeding as

*in*the proof of

*Theorem*22, we also get that for

*some*choice of k depending

*on*d

*and*ε only, we have

*a*PRG where each output string has rK t

*complexity*at most n ...

##
###
Page 1751 of Mathematical Reviews Vol. 52, Issue 5
[page]

1976
*
Mathematical Reviews
*

The author provides

*a*pleasing technique for deriving the*average*number of comparisons required for both*a*successful*and*an unsuccessful search*in**a**random*binary search tree. ...*a*study of the*random*access machine*and*the tandom access acceptor. ...
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