On the Hardness of Permanent

Cai, Jin-Yi; Pavan, A.; Sivakumar, D.

doi:10.1007/3-540-49116-3_8

Cited by 50 publications

(30 citation statements)

References 13 publications

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“…As we do not know how to prove unconditional lower bounds for general circuit classes, a long line of research has focused on hardness amplification. This is the task of transforming worst-case hard functions (or sometimes mildly average-case hard functions) into averagecase hard functions [Yao1,Lip,BF,BFL,BFNW,Imp,GNW,FL,IW1,IW2,CPS,STV,TV,SU1,Tre1,O'D,Vio1,Tre3,HVV,SU2,GK,IJK,IJKW,GG]. This research was largely successful in its goal.…”

Section: Introductionmentioning

confidence: 99%

Hardness Amplification Proofs Require Majority

Shaltiel¹,

Viola²

2010

SIAM J. Comput.

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Hardness amplification is the fundamental task of converting a δ-hard function f : {0, 1} n → {0, 1} into a (1/2 − )-hard function Amp(f ), where f is γ-hard if small circuits fail to compute f on at least a γ fraction of the inputs. Typically, , δ are small (and δ = 2 −k captures the case where f is worst-case hard). Achieving = 1/n ω(1) is a prerequisite for cryptography and most pseudorandom-generator constructions.In this paper we study the complexity of black-box 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 + 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 non-adaptive 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/ bits. In particular, these circuits have large depth when ≤ 1/poly log n.2. The circuits in D must make Ω log(1/δ)/ 2 oracle queries.Both our bounds on the depth and on the number of queries are tight up to constant factors. Our results explain why hardness amplification techniques have failed to transform known lower bounds against constant-depth circuit classes into strong average-case lower bounds. When coupled with the celebrated "Natural Proofs" result by Razborov and Rudich (J. CSS '97) and the pseudorandom functions by Naor and Reingold (J. ACM '04), our results show that standard techniques for hardness amplification can only be applied to those circuit classes for which standard techniques cannot prove circuit lower bounds.Our results reveal a contrast between Yao's XOR Lemmahere Amp(f ) is non-Boolean). Our results (1) and (2) apply to Yao's XOR lemma, whereas known proofs of the direct-product lemma violate both (1) and (2).One of our contributions is a new technique to handle "non-uniform" reductions, i.e. the case when D contains many circuits. * ronen@cs.haifa.ac.il.

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Section: Introductionmentioning

confidence: 99%

Hardness Amplification Proofs Require Majority

Shaltiel¹,

Viola²

2010

SIAM J. Comput.

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“…While this has been accomplished for high complexity classes such #P and EX P (e.g. [17,3,2,7,6,23,25,26]), it remains a major open question for N P. In fact, there are results showing that such connections for N P are unlikely to be provable using the same kinds of techniques used for the high complexity classes [8,26,5].…”

Section: Introductionmentioning

confidence: 99%

Using nondeterminism to amplify hardness

Healy

Vadhan

Viola

2004

Proceedings of the Thirty-Sixth Annual ACM Symposium on Theory of Computing

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“…Likewise, starting with a paper of Lipton, researchers have studied the complexity of computing the permanent (exactly) for many matrices. For example, given an algorithm that computes the permanent exactly for 1/poly(n) fraction of all matrices X over a finite field GF (p) (where p is a sufficiently large prime), one can use self-correction procedures for univariate polynomials [7,8,9] to again obtain efficient randomized algorithms for #P-hard problems.…”

Section: Introductionmentioning

confidence: 99%

Testing Permanent Oracles – Revisited

Arora

Bhattacharyya

Manokaran

et al. 2012

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

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Abstract. Suppose we are given an oracle that claims to approximate the permanent for most matrices X, where X is chosen from the Gaussian ensemble (the matrix entries are i.i.d. univariate complex Gaussians). Can we test that the oracle satisfies this claim? This paper gives a polynomial-time algorithm for the task. The oracle-testing problem is of interest because a recent paper of Aaronson and Arkhipov showed that if there is a polynomial-time algorithm for simulating boson-boson interactions in quantum mechanics, then an approximation oracle for the permanent (of the type described above) exists in BPP NP . Since computing the permanent of even 0/1 matrices is #P-complete, this seems to demonstrate more computational power in quantum mechanics than Shor's factoring algorithm does. However, unlike factoring, which is in NP, it was unclear previously how to test the correctness of an approximation oracle for the permanent, and this is the contribution of the paper. The technical difficulty overcome here is that univariate polynomial selfcorrection, which underlies similar oracle-testing algorithms for permanent over finite fields -and whose discovery led to a revolution in complexity theory-does not seem to generalize to complex (or even, real) numbers. We believe that this tester will motivate further progress on understanding the permanent of Gaussian matrices.

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On the Hardness of Permanent

Cited by 50 publications

References 13 publications

Hardness Amplification Proofs Require Majority

Hardness Amplification Proofs Require Majority

Using nondeterminism to amplify hardness

Testing Permanent Oracles – Revisited

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