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Hardness amplification proofs require majority
 In Proceedings of the 40th Annual ACM Symposium on the Theory of Computing (STOC
, 2008
"... Hardness amplification is the fundamental task of converting a δhard function f: {0, 1} n → {0, 1} into a (1/2 − ɛ)hard function Amp(f), where f is γhard if small circuits fail to compute f on at least a γ fraction of the inputs. Typically, ɛ, δ are small (and δ = 2 −k captures the case where f i ..."
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Hardness amplification is the fundamental task of converting a δhard function f: {0, 1} n → {0, 1} into a (1/2 − ɛ)hard function Amp(f), where f is γhard if small circuits fail to compute f on at least a γ fraction of the inputs. Typically, ɛ, δ are small (and δ = 2 −k captures the case where f is worstcase hard). Achieving ɛ = 1/n ω(1) is a prerequisite for cryptography and most pseudorandomgenerator constructions. In this paper we study the complexity of blackbox proofs of hardness amplification. A class of circuits D proves a hardness amplification result if for any function h that agrees with Amp(f) on a 1/2 + ɛ fraction of the inputs there exists an oracle circuit D ∈ D such that D h agrees with f on a 1 − δ fraction of the inputs. We focus on the case where every D ∈ D makes nonadaptive queries to h. This setting captures most hardness amplification techniques. We prove two main results: 1. The circuits in D “can be used ” to compute the majority function on 1/ɛ bits. In particular, these circuits have large depth when ɛ ≤ 1/poly log n. 2. The circuits in D must make Ω � log(1/δ)/ɛ 2 � oracle queries. Both our bounds on the depth and on the number of queries are tight up to constant factors.
Noiseresilient group testing: Limitations and constructions
 In Proceedings of 17th International Symposium on Fundamentals of Computation Theory (FCT
, 2009
"... We study combinatorial group testing schemes for learning dsparse boolean vectors using highly unreliable disjunctive measurements. We consider an adversarial noise model that only limits the number of false observations, and show that any noiseresilient scheme in this model can only approximately ..."
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Cited by 7 (2 self)
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We study combinatorial group testing schemes for learning dsparse boolean vectors using highly unreliable disjunctive measurements. We consider an adversarial noise model that only limits the number of false observations, and show that any noiseresilient scheme in this model can only approximately reconstruct the sparse vector. On the positive side, we take this barrier to our advantage and show that approximate reconstruction (within a satisfactory degree of approximation) allows us to break the information theoretic lower bound of ˜ Ω(d 2 log n) that is known for exact reconstruction of dsparse vectors of length n via nonadaptive measurements, by a multiplicative factor ˜ Ω(d). Specifically, we give simple randomized constructions of nonadaptive measurement schemes, with m = O(d log n) measurements, that allow efficient reconstruction of dsparse vectors up to O(d) false positives even in the presence of δm false positives and O(m/d) false negatives within the measurement outcomes, for any constant δ < 1. We show that, information theoretically, none of these parameters can be substantially improved without dramatically affecting the others. Furthermore, we obtain several explicit constructions, in particular one matching the randomized tradeoff but using m = O(d 1+o(1) log n) measurements. We also obtain explicit constructions that allow fast reconstruction in time poly(m), which would be sublinear in n for sufficiently sparse vectors. The main tool used in our construction is the listdecoding view of randomness condensers and extractors. An immediate consequence of our result is an adaptive scheme that runs in only two nonadaptive rounds and exactly reconstructs any dsparse vector using a total O(d log n) measurements, a task that would be impossible in one round and fairly easy in O(log(n/d)) rounds.
Perturbation Codes
"... Abstract — We present a new family of codes with good asymptotic properties. These codes are constructed from simple old codes using a new perturbation operator that we introduce. We provide an error reduction algorithm for these codes that uses only elementary operations with small precision. We al ..."
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Abstract — We present a new family of codes with good asymptotic properties. These codes are constructed from simple old codes using a new perturbation operator that we introduce. We provide an error reduction algorithm for these codes that uses only elementary operations with small precision. We also present a soft error reduction algorithm for the expander based codes of AlonBruckNaorNaorRoth when concatenated with any binary code. I.
Query Complexity in Errorless Hardness Amplification
, 2010
"... An errorless circuit for a boolean function is one that outputs the correct answer or “don’t know ” on each input (and never outputs the wrong answer). The goal of errorless hardness amplification is to show that if f has no size s errorless circuit that outputs “don’t know ” on at most a δ fraction ..."
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An errorless circuit for a boolean function is one that outputs the correct answer or “don’t know ” on each input (and never outputs the wrong answer). The goal of errorless hardness amplification is to show that if f has no size s errorless circuit that outputs “don’t know ” on at most a δ fraction of inputs, then some f ′ related to f has no size s ′ errorless circuit that outputs “don’t know ” on at most a 1 − ǫ fraction of inputs. Thus the hardness is “amplified” from δ to 1 −ǫ. Unfortunately, this amplification comes at the cost of a loss in circuit size. This is because such results are proven by reductions which show that any size s ′ errorless circuit for f ′ that outputs “don’t know ” on at most a 1 − ǫ fraction of inputs could be used to construct a size s errorless circuit for f that outputs “don’t know ” on at most a δ fraction of inputs. If the reduction makes q queries to the hypothesized errorless circuit for f ′, then plugging in a size s ′ circuit yields a circuit of size ≥ qs ′, and thus we must have s ′ ≤ s/q. Hence it is desirable to keep the query complexity to a minimum. The first results on errorless hardness amplification were obtained by Bogdanov and Safra. They achieved query complexity O ( ( 1 1 δ log ǫ)2 · 1
A Status Report on the P versus NP Question
"... We survey some of the history of the most famous open question in computing: the P versus NP question. We summarize some of the progress that has been made to date, and assess the current situation. ..."
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We survey some of the history of the most famous open question in computing: the P versus NP question. We summarize some of the progress that has been made to date, and assess the current situation.