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53
Superpolynomial lower bounds for monotone span programs
, 1996
"... In this paper we obtain the first superpolynomial lower bounds for monotone span programs computing explicit functions. The best previous lower bound was Ω(n 5/2) by Beimel, Gál, Paterson [BGP]; our proof exploits a general combinatorial lower bound criterion from that paper. Our lower bounds are ba ..."
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Cited by 48 (6 self)
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In this paper we obtain the first superpolynomial lower bounds for monotone span programs computing explicit functions. The best previous lower bound was Ω(n 5/2) by Beimel, Gál, Paterson [BGP]; our proof exploits a general combinatorial lower bound criterion from that paper. Our lower bounds are based on an analysis of Paleytype bipartite graphs via Weil’s character sum estimates. We prove an n Ω(log n / log log n) lower bound for the size of monotone span programs for the clique problem. Our results give the first superpolynomial lower bounds for linear secret sharing schemes. We demonstrate the surprising power of monotone span programs by exhibiting a function computable in this model in linear size while requiring superpolynomial size monotone circuits and exponential size monotone formulae. We also show that the perfect matching function can be computed by polynomial size (nonmonotone) span programs over arbitrary fields.
From secrecy to soundness: efficient verification via secure computation
 In Proceedings of the 37th international colloquium conference on Automata, languages and programming
, 2010
"... Abstract. We study the problem of verifiable computation (VC) in which a computationally weak client wishes to delegate the computation of a function f on an input x to a computationally strong but untrusted server. We present new general approaches for constructing VC protocols, as well as solving ..."
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Cited by 46 (3 self)
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Abstract. We study the problem of verifiable computation (VC) in which a computationally weak client wishes to delegate the computation of a function f on an input x to a computationally strong but untrusted server. We present new general approaches for constructing VC protocols, as well as solving the related problems of program checking and selfcorrecting. The new approaches reduce the task of verifiable computation to suitable variants of secure multiparty computation (MPC) protocols. In particular, we show how to efficiently convert the secrecy property of MPC protocols into soundness of a VC protocol via the use of a message authentication code (MAC). The new connections allow us to apply results from the area of MPC towards simplifying, unifying, and improving over previous results on VC and related problems. In particular, we obtain the following concrete applications: (1) The first VC protocols for arithmetic computations which only make a blackbox use of the underlying field or ring; (2) a noninteractive VC protocol for boolean circuits in the preprocessing model, conceptually simplifying and improving the online complexity of a recent protocol of Gennaro et al. (Cryptology ePrint Archive: Report 2009/547); (3) NC0 selfcorrectors for complete languages in the complexity class NC1 and various logspace classes, strengthening previous AC0 correctors of Goldwasser et al. (STOC 2008). 1
On the Hardness of Graph Isomorphism
 SIAM J. COMPUT
"... We show that the graph isomorphism problem is hard under DLOGTIME uniform AC0 manyone reductions for the complexity classes NL, PL (probabilistic logarithmic space) for every logarithmic space modular class ModkL and for the class DET of problems NC¹ reducible to the determinant. These are the stro ..."
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Cited by 44 (1 self)
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We show that the graph isomorphism problem is hard under DLOGTIME uniform AC0 manyone reductions for the complexity classes NL, PL (probabilistic logarithmic space) for every logarithmic space modular class ModkL and for the class DET of problems NC¹ reducible to the determinant. These are the strongest known hardness results for the graph isomorphism problem and imply a randomized logarithmic space reduction from the perfect matching problem to graph isomorphism. We also investigate hardness results for the graph automorphism problem.
The Complexity of Matrix Rank and Feasible Systems of Linear Equations
"... We characterize the complexity of some natural and important problems in linear algebra. In particular, we identify natural complexity classes for which the problems of (a) determining if a system of linear equations is feasible and (b) computing the rank of an integer matrix, (as well as other p ..."
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Cited by 41 (8 self)
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We characterize the complexity of some natural and important problems in linear algebra. In particular, we identify natural complexity classes for which the problems of (a) determining if a system of linear equations is feasible and (b) computing the rank of an integer matrix, (as well as other problems), are complete under logspace reductions. As an important
Universal algebra and hardness results for constraint satisfaction problems
, 2007
"... Abstract. We present algebraic conditions on constraint languages Γ that ensure the hardness of the constraint satisfaction problem CSP(Γ) for complexity classes L, NL, P, NP and ModpL. These criteria also give nonexpressibility results for various restrictions of Datalog. Furthermore, we show tha ..."
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Cited by 36 (7 self)
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Abstract. We present algebraic conditions on constraint languages Γ that ensure the hardness of the constraint satisfaction problem CSP(Γ) for complexity classes L, NL, P, NP and ModpL. These criteria also give nonexpressibility results for various restrictions of Datalog. Furthermore, we show that if CSP(Γ) is not firstorder definable then it is Lhard. Our proofs rely on tame congruence theory and on a finegrain analysis of the complexity of reductions used in the algebraic study of CSP. The results pave the way for a refinement of the dichotomy conjecture stating that each CSP(Γ) lies in P or is NPcomplete and they match the recent classification of [2] for Boolean CSP. We also infer a partial classification theorem for the complexity of CSP(Γ) when the associated algebra of Γ is the full idempotent reduct of a preprimal algebra. Constraint satisfaction problems (CSP) provide a unifying framework to study various computational problems arising naturally in artificial intelligence, combinatorial optimization, graph homomorphisms and database theory. An in
Isolation, Matching, and Counting: Uniform and Nonuniform Upper Bounds
 Journal of Computer and System Sciences
, 1998
"... We show that the perfect matching problem is in the complexity class SPL (in the nonuniform setting). This provides a better upper bound on the complexity of the matching problem, as well as providing motivation for studying the complexity class SPL. Using similar techniques, we show that counting t ..."
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Cited by 30 (6 self)
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We show that the perfect matching problem is in the complexity class SPL (in the nonuniform setting). This provides a better upper bound on the complexity of the matching problem, as well as providing motivation for studying the complexity class SPL. Using similar techniques, we show that counting the number of accepting paths of a nondeterministic logspace machine can be done in NL/poly, if the number of paths is small. This clarifies the complexity of the class LogFew (defined and studied in [BDHM91]). Using derandomization techniques, we then improve this to show that this counting problem is in NL. Determining if our other theorems hold in the uniform setting remains an The material in this paper appeared in preliminary form in papers in the Proceedings of the IEEE Conference on Computational Complexity, 1998, and in the Proceedings of the Workshop on Randomized Algorithms, Brno, 1998. y Supported in part by NSF grants CCR9509603 and CCR9734918. z Supported in part by the ...
Arithmetic circuits and counting complexity classes
 In Complexity of Computations and Proofs,J.Krajíček, Ed. Quaderni di Matematica
"... Arithmetic circuits are the focus of renewed attention in the complexity theory community. It is easy to list a few of the reasons for the increased interest: • Innovative work by Kabanets and Impagliazzo [KI03] shows that, in ..."
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Cited by 27 (5 self)
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Arithmetic circuits are the focus of renewed attention in the complexity theory community. It is easy to list a few of the reasons for the increased interest: • Innovative work by Kabanets and Impagliazzo [KI03] shows that, in
Logspace and Logtime Leaf Languages
, 1996
"... The computation tree of a nondeterministic machine M with input x gives rise to a leaf string formed by concatenating the outcomes of all the computations in the tree in lexicographical order. We may characterize problems by considering, for a particular "leaf language" Y , the set of all ..."
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Cited by 25 (2 self)
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The computation tree of a nondeterministic machine M with input x gives rise to a leaf string formed by concatenating the outcomes of all the computations in the tree in lexicographical order. We may characterize problems by considering, for a particular "leaf language" Y , the set of all x for which the leaf string of M is contained in Y . In this way, in the context of polynomial time computation, leaf languages were shown to capture many complexity classes. In this paper, we study the expressibility of the leaf language mechanism in the contexts of logarithmic space and of logarithmic time computation. We show that logspace leaf languages yield a much finer classification scheme for complexity classes than polynomial time leaf languages, capturing also many classes within P. In contrast, logtime leaf languages basically behave like logtime reducibilities. Both cases are more subtle to handle than the polynomial time case. We also raise the issue of balanced versus nonbalanced comp...
Unambiguity and Fewness for Logarithmic Space
, 1991
"... We consider various types of unambiguity for logarithmic space bounded Turing machines and polynomial time bounded log space auxiliary pushdown automata. In particular, we introduce the notions of (general), reach, and strong unambiguity. We demonstrate that closure under complement of unambiguo ..."
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Cited by 25 (6 self)
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We consider various types of unambiguity for logarithmic space bounded Turing machines and polynomial time bounded log space auxiliary pushdown automata. In particular, we introduce the notions of (general), reach, and strong unambiguity. We demonstrate that closure under complement of unambiguous classes implies equivalence of unambiguity and "unambiguous fewness". This, as we will show, applies in the cases of reach and strong unambiguity for logspace. Among the many relations we exhibit, we show that the unambiguous linear contextfree languages, which are not known to be contained in LOGSPACE, nevertheless are contained in strongly unambiguous logspace, and, consequently, in LOGDCFL. In fact, this turns out to be the case for all logspace languages with reach unambiguous fewness. In addition, we show that general unambiguity and fewness of logspace classes can be simulated by reach unambiguity and fewness of auxiliary pushdown automata. 1 Introduction Although the pow...