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18
Which Problems Have Strongly Exponential Complexity?
 Journal of Computer and System Sciences
, 1998
"... For several NPcomplete problems, there have been a progression of better but still exponential algorithms. In this paper, we address the relative likelihood of subexponential algorithms for these problems. We introduce a generalized reduction which we call SubExponential Reduction Family (SERF) t ..."
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Cited by 249 (9 self)
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For several NPcomplete problems, there have been a progression of better but still exponential algorithms. In this paper, we address the relative likelihood of subexponential algorithms for these problems. We introduce a generalized reduction which we call SubExponential Reduction Family (SERF) that preserves subexponential complexity. We show that CircuitSAT is SERFcomplete for all NPsearch problems, and that for any fixed k, kSAT, kColorability, kSet Cover, Independent Set, Clique, Vertex Cover, are SERFcomplete for the class SNP of search problems expressible by second order existential formulas whose first order part is universal. In particular, subexponential complexity for any one of the above problems implies the same for all others. We also look at the issue of proving strongly exponential lower bounds for AC 0 ; that is, bounds of the form 2 \Omega\Gamma n) . This problem is even open for depth3 circuits. In fact, such a bound for depth3 circuits with even l...
An Improved Exponentialtime Algorithm for kSAT
, 1998
"... We propose and analyze a simple new randomized algorithm, called ResolveSat, for finding satisfying assignments of Boolean formulas in conjunctive normal form. The algorithm consists of two stages: a preprocessing stage in which resolution is applied to enlarge the set of clauses of the formula, ..."
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Cited by 116 (7 self)
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We propose and analyze a simple new randomized algorithm, called ResolveSat, for finding satisfying assignments of Boolean formulas in conjunctive normal form. The algorithm consists of two stages: a preprocessing stage in which resolution is applied to enlarge the set of clauses of the formula, followed by a search stage that uses a simple randomized greedy procedure to look for a satisfying assignment. We show that, for each k, the running time of ResolveSat on a kCNF formula is significantly better than 2 n , even in the worst case. In particular, we show that the algorithm finds a satisfying assignment of a general satisfiable 3CNF in time O(2 :448n ) with high probability; where the best previous algorithm [13] has running time O(2 :562n ). We obtain a better upper bound of 2 (2 ln 2\Gamma1)n+o(n) = O(2 0:387n ) for 3CNF that have exactly one satisfying assignment (unique kSAT). For each k, the bounds for general kCNF are the best currently known for ...
Relativizable And Nonrelativizable Theorems In The Polynomial Theory Of Algorithms
 In Russian
, 1993
"... . Starting with the paper of Baker, Gill and Solovay [BGS 75] in complexity theory, many results have been proved which separate certain relativized complexity classes or show that they have no complete language. All results of this kind were, in fact, based on lower bounds for boolean decision tree ..."
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Cited by 38 (0 self)
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. Starting with the paper of Baker, Gill and Solovay [BGS 75] in complexity theory, many results have been proved which separate certain relativized complexity classes or show that they have no complete language. All results of this kind were, in fact, based on lower bounds for boolean decision trees of a certain type or for machines with polylogarithmic restrictions on time. The following question arises: Are these methods of proving "relativized" results universal? In the first part of the present paper we propose a general framework in which assertions of universality of this kind may be formulated and proved as convenient criteria. Using these criteria we obtain, as easy consequences of the known results on boolean decision trees, some new "relativized" results and new proofs of some known results. In the second part of the present paper we apply these general criteria to many particular cases. For example, for many of the complexity classes studied in the literature all relativiza...
Exponential lower bounds for depth 3 boolean circuits
 Preliminary version in 29th annual ACM Symposium on Theory of Computing, 96–91
, 2000
"... ..."
Affine Projections of Symmetric Polynomials
 In Proc. 16th Annual IEEE Conference on Computational Complexity
, 2001
"... In this paper we introduce a new model for computing polynomials  a depth 2 circuit with a symmetric gate at the top and plus gates at the bottom, i.e the circuit computes a symmetric function in linear functions  S d m (` 1 ; ` 2 ; :::; ` m ) (S d m is the d'th elementary symmetric polyno ..."
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Cited by 7 (1 self)
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In this paper we introduce a new model for computing polynomials  a depth 2 circuit with a symmetric gate at the top and plus gates at the bottom, i.e the circuit computes a symmetric function in linear functions  S d m (` 1 ; ` 2 ; :::; ` m ) (S d m is the d'th elementary symmetric polynomial in m variables, and the ` i 's are linear functions). We refer to this model as the symmetric model. This new model is related to standard models of arithmetic circuits, especially to depth 3 circuits. In particular we show that in order to improve the results of [19], i.e to prove superquadratic lower bounds for depth 3 circuits, one must first prove a superlinear lower bound for the symmetric model. We prove two nontrivial linear lower bounds for our model. The first lower bound is for computing the determinant, and the second is for computing the sum of two monomials. The main technical contribution relates the maximal dimension of linear subspaces on which S d m vanishes, and lower bounds to the symmetric model. In particular we show that an answer of the following problem (which is very natural, and of independent interest) will imply lower bounds on symmetric circuits for many polynomials: "What is the maximal dimension of a linear subspace of C m , on which S d m vanishes ?" We give two partial solutions to the problem above, each enables us to prove a different lower bound. Using our techniques we also prove quadratic lower bounds for depth 3 circuits computing the elementary symmetric polynomials of degree n (where 0 < < 1 is a constant), thus extending the result of [19]. These are the best lower bounds known for depth 3 circuits over fields of characteristic zero. 1.
On the correlation between parity and modular polynomials
 Proceedings of the 31st International Symposium on Mathematical Foundations of Computer Science
, 2006
"... Abstract. We consider the problem of bounding the correlation between parity and modular polynomials over Zq, for arbitrary odd integer q ≥ 3. We prove exponentially small upper bounds for classes of polynomials with certain linear algebraic properties. As a corollary, we obtain exponential lower bo ..."
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Cited by 6 (2 self)
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Abstract. We consider the problem of bounding the correlation between parity and modular polynomials over Zq, for arbitrary odd integer q ≥ 3. We prove exponentially small upper bounds for classes of polynomials with certain linear algebraic properties. As a corollary, we obtain exponential lower bounds on the size necessary to compute parity by depth3 circuits of certain form. Our technique is based on a new representation of the correlation using exponential sums. Our results include Goldmann’s result [Go] on the correlation between parity and degree one polynomials as a special case. Our general expression for representing correlation can be used to derive the bounds of Cai, Green, and Thierauf [CGT] for symmetric polynomials, using ideas of the [CGT] proof. The classes of polynomials for which we obtain exponentially small upper bounds include polynomials of large degree and with a large number of terms, that previous techniques did not apply to. 1
THE DOTDEPTH AND THE POLYNOMIAL HIERARCHIES CORRESPOND ON THE DELTA LEVELS
 INTERNATIONAL JOURNAL OF FOUNDATIONS OF COMPUTER SCIENCE
"... It is wellknown that the Σk and Πklevels of the dotdepth hierarchy and the polynomial hierarchy correspond via leaf languages. We extend this correspondence to the ∆klevels of these hierarchies: LeafP (∆L k) = ∆p k. The same methods are used to give evidence against an earlier conjecture of S ..."
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Cited by 5 (1 self)
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It is wellknown that the Σk and Πklevels of the dotdepth hierarchy and the polynomial hierarchy correspond via leaf languages. We extend this correspondence to the ∆klevels of these hierarchies: LeafP (∆L k) = ∆p k. The same methods are used to give evidence against an earlier conjecture of Straubing and Thérien about a leaflanguage upper bound for BPP.
Circuit Lower Bounds à la Kolmogorov
"... In a recent paper, Razborov [Raz93] gave a new combinatorial proof of Hastad's switching lemma [Has89], eliminating the probabilistic argument altogether. In this paper we adapt his proof and propose a Kolmogorov complexitystyle switching lemma, from which we derive the probabilistic switching ..."
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Cited by 4 (1 self)
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In a recent paper, Razborov [Raz93] gave a new combinatorial proof of Hastad's switching lemma [Has89], eliminating the probabilistic argument altogether. In this paper we adapt his proof and propose a Kolmogorov complexitystyle switching lemma, from which we derive the probabilistic switching lemma as well as a Kolmogorov complexitystyle proof of circuit lower bounds for parity. Hastad's switching lemma [Has89] is a prime example of the socalled "restriction" or "bottomup" method used to find lower bounds in circuit complexity. This approach consists of considering circuits from the bottom level (inputs), and showing that restricting the function by fixing some of the inputs does not always force the function to zero or one. In a recent paper [Raz93], Razborov presented a new proof of Hastad's lemma, using a simpler counting argument instead of the usual probabilistic techniques. (See also the paper by Beame [Bea94] for a presentation of this proof, as well as extensions of the ...
A Kolmogorov Complexity proof of Håstad's switching lemma  An Exposition
, 1994
"... We present here a proof of Hastad's switching lemma. The switching lemma has important applications in the proof of lower bounds in circuit complexity as well as other areas of complexity theory. The proof presented here is based on the one presented in [5], expressed in terms of Kolmogorov com ..."
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Cited by 1 (0 self)
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We present here a proof of Hastad's switching lemma. The switching lemma has important applications in the proof of lower bounds in circuit complexity as well as other areas of complexity theory. The proof presented here is based on the one presented in [5], expressed in terms of Kolmogorov complexity by Lance Fortnow. Keywords: computational complexity, theory of computation. Hastad's switching lemma [3] is a prime example of the socalled "restriction " or "bottomup" method used to find lower bounds in circuit complexity. This approach consists of considering circuits from the bottom level (inputs), and showing that restricting the function by fixing some of the inputs does not always force the function to zero or one. Applications of Hasta's lemma include lower bounds on the size of constantdepth, unbounded fanin circuits for parity ([3]), and the construction of oracles for which the polynomialtime hierarchy is infinite ([2, 6]). In [5], Razborov presented a new proof of Hastad...