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A theorem on random polynomials and some consequences in average complexity

F. Cucker, M.-F. Roy
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  ... 
doi:10.1016/s0747-7171(08)80052-1 fatcat:zhiama4bgrgsjjceycsvifsjoi

Some Results on Derandomization

Harry Buhrman, Lance Fortnow, A. Pavan
2004 Theory of Computing Systems  
If 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.  ... 
doi:10.1007/s00224-004-1194-y fatcat:67kyob6puzevhit4ov5dyztx2q

Some Results on Derandomization [chapter]

Harry Buhrman, Lance Fortnow, A. Pavan
2003 Lecture Notes in Computer Science  
If 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.  ... 
doi:10.1007/3-540-36494-3_20 fatcat:g5suojh4fffrvimr5q4fixejle

Average-case intractability vs. worst-case intractability

Johannes Köbler, Rainer Schuler
2004 Information and Computation  
Furthermore, other 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.  ... 
doi:10.1016/j.ic.2003.05.002 fatcat:v7cyr4ycljf4to6d4hh5haqt7e

Average-case intractability vs. worst-case intractability [chapter]

Johannes Köbler, Rainer Schuler
1998 Lecture Notes in Computer Science  
As 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.  ... 
doi:10.1007/bfb0055799 fatcat:x4jqsloj4be2fmn5pdze6yglby

Roots of the derivatives of some random polynomials

André Galligo
2011 Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation - SNC '11  
Our observations show that the sets of real (respectively 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.  ... 
doi:10.1145/2331684.2331703 dblp:conf/snc/Galligo11 fatcat:ar7zvksywvcmleo5ivwmacneu4

Fine-Grained Derandomization: From Problem-Centric to Resource-Centric Complexity

Marco L. Carmosino, Russell Impagliazzo, Manuel Sabin, Michael Wagner
2018 International Colloquium on Automata, Languages and Programming  
problems) 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  ... 
doi:10.4230/lipics.icalp.2018.27 dblp:conf/icalp/CarmosinoIS18 fatcat:e45lnmtgxncpnbnaxkrafnmdou

Average-Case Complexity Versus Approximate Simulation of Commuting Quantum Computations

Michael J. Bremner, Ashley Montanaro, Dan J. Shepherd
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  ... 
doi:10.1103/physrevlett.117.080501 pmid:27588839 fatcat:jjn4gnqlvncitdunlih554uiiq

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

Zhenjian Lu, Igor C. Oliveira, Marius Zimand
2022 arXiv   pre-print
running times of languages that are 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.  ... 
arXiv:2204.08312v1 fatcat:g5y7win7wndgdguem6t5xxzqrq

Pseudorandomness and the Minimum Circuit Size Problem

Rahul Santhanam, Michael Wagner
2020 Innovations in Theoretical Computer Science  
We also show that for 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.  ... 
doi:10.4230/lipics.itcs.2020.68 dblp:conf/innovations/Santhanam20 fatcat:pgatorc5k5hlrknytijhfgjufy


2004 Current Trends in Theoretical Computer Science  
This survey focuses 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  ... 
doi:10.1142/9789812562494_0012 fatcat:znfvduwgrjexpcbcj62ientmvy

Condition Numbers for the Cube. I: Univariate Polynomials and Hypersurfaces [article]

Josué Tonelli-Cueto, Elias Tsigaridas
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.  ... 
arXiv:2006.04423v3 fatcat:qnbw47ivfndvpmtxub2avydel4

Hardness Magnification for Natural Problems

Igor Carboni Oliveira, Rahul Santhanam
2018 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)  
We show that if MCSP[2 √ n ] cannot be solved 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.  ... 
doi:10.1109/focs.2018.00016 dblp:conf/focs/OliveiraS18 fatcat:4youvi7wjfbxzn32ttjuy4c6f4

Pseudodeterministic algorithms and the structure of probabilistic time

Zhenjian Lu, Igor C. Oliveira, Rahul Santhanam
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  ... 
doi:10.1145/3406325.3451085 fatcat:yqh6zf42y5akrflgxvwvianpmm

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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