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Relativized Worlds without Worst-Case to Average-Case Reductions for NP [chapter]

Thomas Watson
2010 Lecture Notes in Computer Science  
We prove that relative to an oracle, there is no worst-case to average-case reduction for NP.  ...  In fact, we prove that relative to an oracle, there is no worst-case to errorless-average-case reduction from NP to BPP NP .  ...  Acknowledgments First and foremost, I thank Luca Trevisan for suggesting the research topic. I thank Luca Trevisan, Dieter van Melkebeek, Ronen Shaltiel, and anonymous reviewers for helpful comments.  ... 
doi:10.1007/978-3-642-15369-3_56 fatcat:6nedksj2j5gnpigavopxrjho34

Relativized Worlds without Worst-Case to Average-Case Reductions for NP

Thomas Watson
2012 ACM Transactions on Computation Theory  
We prove that relative to an oracle, there is no worst-case to average-case reduction for NP.  ...  In fact, we prove that relative to an oracle, there is no worst-case to errorless-average-case reduction from NP to BPP NP .  ...  Acknowledgments First and foremost, I thank Luca Trevisan for suggesting the research topic. I thank Luca Trevisan, Dieter van Melkebeek, Ronen Shaltiel, and anonymous reviewers for helpful comments.  ... 
doi:10.1145/2355580.2355583 fatcat:32ecj6wnlndyvdo7iw6plbn3fq

If NP Languages are Hard on the Worst-Case, Then it is Easy to Find Their Hard Instances

Dan Gutfreund, Ronen Shaltiel, Amnon Ta-Shma
2007 Computational Complexity  
This is the first worst-case to average-case reduction for NP of any kind.  ...  (unless NP is easy in the worst case).  ...  We also thank Salil and Rahul Santhanam for valuable comments on an earlier version of the manuscript. We thank the anonymous referees for helpful comments.  ... 
doi:10.1007/s00037-007-0235-8 fatcat:x2ubi7gojzdlzdrpudwyuvwvd4

New connections between derandomization, worst-case complexity and average-case complexity [article]

Dan Gutfreund, Amnon Ta-Shma
2006 Electronic colloquium on computational complexity  
Thus our techniques show a way to bypass black-box impossibility arguments regarding worst-case to average-case reductions.  ...  There has been a long line of research trying to explain our failure in proving worstcase to average-case reductions within NP [FF93, Vio03, BT03, AGGM06].  ...  Andrej Bogdanov for pointing out to us that Theorem 5.2 follows directly from a generalization of [BT03] (i.e. Theorem 5.1). Previously we had a more complicated and weaker statement. Charlie  ... 
dblp:journals/eccc/GutfreundT06 fatcat:a34uuphxkndbti3rzsrzje2ke4

Structural Average Case Complexity

Rainer Schuler, Tomoyuki Yamakami
1996 Journal of computer and system sciences (Print)  
A new notion of \real polynomial-time computations" is introduced based on average polynomial-time computations for arbitrary distributions from a xed set, and it is used to characterize the worst-case  ...  for Random-NP are Turing self-reducible.  ...  Acknowledgments The authors would like to thank Professor Jos e Balc azar and Professor Uwe S c h oning for organizing workshops in Barcelona and Ulm in 1991 where many discussions on average-case complexity  ... 
doi:10.1006/jcss.1996.0024 fatcat:muvu7uuxiva2rfo2p4ygv7zpdy

Structural average case complexity [chapter]

Rainer Schuler, Tomoyuki Yamakami
1992 Lecture Notes in Computer Science  
A new notion of \real polynomial-time computations" is introduced based on average polynomial-time computations for arbitrary distributions from a xed set, and it is used to characterize the worst-case  ...  for Random-NP are Turing self-reducible.  ...  Acknowledgments The authors would like to thank Professor Jos e Balc azar and Professor Uwe S c h oning for organizing workshops in Barcelona and Ulm in 1991 where many discussions on average-case complexity  ... 
doi:10.1007/3-540-56287-7_100 fatcat:rcfbjdt24rfjpav36znuhiyfke

Finding Pessiland [chapter]

Hoeteck Wee
2006 Lecture Notes in Computer Science  
Introduction Pessiland, coined by Impagliazzo [I95], is a world in which there are hard-on-average languages in NP but no one-way functions.  ...  For 2-round argument systems, it is known that a relaxation of hard-on-average languages in NP is necessary [W05] (also, Appendix A.2).  ...  In addition, I thank Lance Fortnow and Russell Impagliazzo for pointing out previous constructions of Pessiland, and the anonymous referees for their helpful and constructive feedback.  ... 
doi:10.1007/11681878_22 fatcat:rilppn3n3zfb3ikgtiq74ljucu

Towards Non-Black-Box Lower Bounds in Cryptography [chapter]

Rafael Pass, Wei-Lung Dustin Tseng, Muthuramakrishnan Venkitasubramaniam
2011 Lecture Notes in Computer Science  
We consider average-case strengthenings of the traditional assumption that coNP is not contained in AM.  ...  ., one-way permutations, collision-resistant hash-functions, constant-round statistically hiding commitments, and constant-round black-box zero-knowledge proofs for NP) from one-way functions, assuming  ...  Acknowledgements We are thankful to the anonymous TCC-reviewers for their insightful comments.  ... 
doi:10.1007/978-3-642-19571-6_35 fatcat:gd4qwcndgzfihe7xnuv373p354

Structure vs. Hardness Through the Obfuscation Lens [chapter]

Nir Bitansky, Akshay Degwekar, Vinod Vaikuntanathan
2017 Lecture Notes in Computer Science  
There is no fully black-box reduction of any (even worst-case) hard problem in SZK to IO and OWPs. Corollary 1.2 (from [SW14, Wat15], Informal).  ...  The existence of one-way permutations (OWPs) requires a hard problem in NP ∩ coNP [Bra79]; the same holds for restricted cases of public-key encryption schemes satisfying specific structural properties  ...  We also thank the anonymous FOCS'16 reviewers for their valuable comments.  ... 
doi:10.1007/978-3-319-63688-7_23 fatcat:tw76c3gqnvetvfburf4ttppsza

On the Existence of Weak One-Way Functions [article]

Stefan Rass
2017 arXiv   pre-print
of the decision problem (language) L_0 (however, without any auxiliary information that could ease the decision like a polynomial witness).  ...  well-known discrete time-hierarchy theorem from complexity theory), we construct another intractable decision problem L⊆{0,1}^* that has its words scattered across {0,1}^ℓ at a relative frequency p(ℓ), for  ...  Acknowledgment The author is indebted to Sandra König from the Austrian Institute of Technology, for spotting some errors, as well as to Patrick Horster, for valuable discussions about earlier versions  ... 
arXiv:1609.01575v3 fatcat:va2sg7kmv5c3jjk4dvleta6bb4

The status of the P versus NP problem

Lance Fortnow
2009 Communications of the ACM  
Acknowledgments Thanks to Rahul Santhanam for many useful discussions and comments. Josh Grochow wrote an early draft of Section 9. The anonymous referees provided critical advice.  ...  Whether similar worst-to-average reductions hold for NP-complete sets is an important open problem.  ...  Heuristics and Average-Case Complexity The study of NP-completeness focuses on how algorithms perform on the worst possible inputs.  ... 
doi:10.1145/1562164.1562186 fatcat:gxquaagohnc7jdxdhaaexzof44

An Atypical Survey of Typical-Case Heuristic Algorithms [article]

Lane A. Hemaspaandra, Ryan Williams
2012 arXiv   pre-print
How close can heuristic algorithms get to always giving the right answer, without inducing seismic complexity-theoretic consequences?  ...  Heuristic approaches often do so well that they seem to pretty much always give the right answer.  ...  Acknowledgments We are grateful to Paul Beame for providing helpful SAT references. References  ... 
arXiv:1210.8099v1 fatcat:g52yqzbgtjd4bb2jypg6fw6txa

One-way Functions and Partial MCSP [article]

Eric Allender, Mahdi Cheraghchi, Dimitrios Myrisiotis, Harsha Tirumala, Ilya Volkovich
2021 Electronic colloquium on computational complexity  
It remained an open problem to establish such an equivalence for the average-case hardness of some natural NP-complete problem.  ...  Thus the existence of OWFs is inextricably linked to the average-case hardness of this NP-complete problem.  ...  Acknowledgements We would like to thank Russell Impagliazzo for explaining his work [IL90] to us, and Ján Pich and Ninad Rajgopal for illuminating discussions.  ... 
dblp:journals/eccc/AllenderCMTV21 fatcat:og52i7xvhbbadpwpz7ex7r2egq

The P versus NP Problem [article]

Rakesh Dube
2010 arXiv   pre-print
See http://gauss.claymath.org:8888/millennium/P_vs_NP/pvsnp.pdf for the original text.  ...  This article was plagiarized directly from Stephen Cook's description of the problem for the Clay Mathematics Institute.  ...  at least steps in the worst case, for some .  ... 
arXiv:1001.3816v2 fatcat:ghl35v6tdjfkdmvqgl5g6cb53i

The NP-completeness column

David S. Johnson
2007 ACM Transactions on Algorithms  
Garey and I used in our book "Computers and Intractability: A Guide to the Theory of NP-Completeness," W. H. Freeman & Co., New York, 1979, hereinafter referred to as " [G&J]."  ...  This is the 26th edition of a column that covers new developments in the theory of NP-completeness. The presentation is modeled on that which M. R.  ...  Each could turn out to be in P without implying that NP ∩ co-NP equals P.  ... 
doi:10.1145/1240233.1240247 fatcat:2rnecxy6ufcihmva53od5i6234
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