Results 1  10
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142
Monotone Complexity
, 1990
"... We give a general complexity classification scheme for monotone computation, including monotone spacebounded and Turing machine models not previously considered. We propose monotone complexity classes including mAC i , mNC i , mLOGCFL, mBWBP , mL, mNL, mP , mBPP and mNP . We define a simple ..."
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Cited by 2429 (12 self)
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We give a general complexity classification scheme for monotone computation, including monotone spacebounded and Turing machine models not previously considered. We propose monotone complexity classes including mAC i , mNC i , mLOGCFL, mBWBP , mL, mNL, mP , mBPP and mNP . We define a simple notion of monotone reducibility and exhibit complete problems. This provides a framework for stating existing results and asking new questions. We show that mNL (monotone nondeterministic logspace) is not closed under complementation, in contrast to Immerman's and Szelepcs 'enyi's nonmonotone result [Imm88, Sze87] that NL = coNL; this is a simple extension of the monotone circuit depth lower bound of Karchmer and Wigderson [KW90] for stconnectivity. We also consider mBWBP (monotone bounded width branching programs) and study the question of whether mBWBP is properly contained in mNC 1 , motivated by Barrington's result [Bar89] that BWBP = NC 1 . Although we cannot answer t...
Almost Optimal Lower Bounds for Small Depth Circuits
 RANDOMNESS AND COMPUTATION
, 1989
"... We give improved lower bounds for the size of small depth circuits computing several functions. In particular we prove almost optimal lower bounds for the size of parity circuits. Further we show that there are functions computable in polynomial size and depth k but requires exponential size when ..."
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Cited by 235 (8 self)
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We give improved lower bounds for the size of small depth circuits computing several functions. In particular we prove almost optimal lower bounds for the size of parity circuits. Further we show that there are functions computable in polynomial size and depth k but requires exponential size when the depth is restricted to k1. Our main lemma which is of independent interest states that by using a random restriction we can convert an AND of small ORs to an OR of small ANDs and conversely.
The NPcompleteness column: an ongoing guide
 Journal of Algorithms
, 1985
"... This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & ..."
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Cited by 196 (0 self)
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This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & Co., New York, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, crossreferences will be given to that book and the list of problems (NPcomplete and harder) presented there. Readers who have results they would like mentioned (NPhardness, PSPACEhardness, polynomialtimesolvability, etc.) or open problems they would like publicized, should
Lower Bounds for Resolution and Cutting Plane Proofs and Monotone Computations
, 1997
"... We prove an exponential lower bound on the length of cutting plane proofs. The proof uses an extension of a lower bound for monotone circuits to circuits which compute with real numbers and use nondecreasing functions as gates. The latter result is of independent interest, since, in particular, i ..."
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Cited by 139 (5 self)
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We prove an exponential lower bound on the length of cutting plane proofs. The proof uses an extension of a lower bound for monotone circuits to circuits which compute with real numbers and use nondecreasing functions as gates. The latter result is of independent interest, since, in particular, it implies an exponential lower bound for some arithmetic circuits.
The monotone circuit complexity of Boolean functions
 COMBINATORICA
, 1987
"... Recently, Razborov obtained superpolynomial lower bounds for monotone circuits that lect cliques in graphs. In particular, Razborov showed that detecting cliques of size s in a graph dh m vertices requires monotone circuits of size.Q(m'/(log m) ~') for fixed s, and size rn ao°~') for ..."
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Cited by 130 (3 self)
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Recently, Razborov obtained superpolynomial lower bounds for monotone circuits that lect cliques in graphs. In particular, Razborov showed that detecting cliques of size s in a graph dh m vertices requires monotone circuits of size.Q(m'/(log m) ~') for fixed s, and size rn ao°~') for,.:[log ml4J. In this paper we modify the arguments of Razborov to obtain exponential lower bounds for monotone circuits. In particular, detecting cliques of size (1/4) (m/log m) ~'/a requires monotone circuits f size exp (£2((m/log m)~/:~)). For fixed s, any monotone circuit that detects cliques of size s requires 'm'/(log m)') AND gates. We show that even a very rough approximation of the maximum clique e of a graph requires superpolynomial size monotone circuits, and give lower bounds for some net Boolean functions. Our best lower bound fi~r an NP function of n variables is exp (f2(n w4. (log n)~/~)), improving a recent result of exp (f2(nws')) due to Andreev.
Interpolation Theorems, Lower Bounds for Proof Systems, and Independence Results for Bounded Arithmetic
"... A proof of the (propositional) Craig interpolation theorem for cutfree sequent calculus yields that a sequent with a cutfree proof (or with a proof with cutformulas of restricted form; in particular, with only analytic cuts) with k inferences has an interpolant whose circuitsize is at most k. We ..."
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Cited by 88 (2 self)
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A proof of the (propositional) Craig interpolation theorem for cutfree sequent calculus yields that a sequent with a cutfree proof (or with a proof with cutformulas of restricted form; in particular, with only analytic cuts) with k inferences has an interpolant whose circuitsize is at most k. We give a new proof of the interpolation theorem based on a communication complexity approach which allows a similar estimate for a larger class of proofs. We derive from it several corollaries: 1. Feasible interpolation theorems for the following proof systems: (a) resolution. (b) a subsystem of LK corresponding to the bounded arithmetic theory S 2 2 (ff). (c) linear equational calculus. (d) cutting planes. 2. New proofs of the exponential lower bounds (for new formulas) (a) for resolution ([15]). (b) for the cutting planes proof system with coefficients written in unary ([4]). 3. An alternative proof of the independence result of [43] concerning the provability of circuitsize lower bounds ...
Lower Bounds for the Size of Circuits of Bounded Depth in Basis
, 1986
"... this paper, we consider circuits of bounded depth in the basis f; \Phig. ..."
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Cited by 79 (0 self)
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this paper, we consider circuits of bounded depth in the basis f; \Phig.
Lower Bounds for Cutting Planes Proofs with Small Coefficients
, 1995
"... We consider smallweight Cutting Planes (CP ) proofs; that is, Cutting Planes (CP ) proofs with coefficients up to P oly(n). We use the well known lower bounds for monotone complexity to prove an exponential lower bound for the length of CP proofs, for a family of tautologies based on the cl ..."
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Cited by 76 (17 self)
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We consider smallweight Cutting Planes (CP ) proofs; that is, Cutting Planes (CP ) proofs with coefficients up to P oly(n). We use the well known lower bounds for monotone complexity to prove an exponential lower bound for the length of CP proofs, for a family of tautologies based on the clique function. Because Resolution is a special case of smallweight CP , our method also gives a new and simpler exponential lower bound for Resolution. We also prove the following two theorems : (1) Treelike CP proofs cannot polynomially simulate nontreelike CP proofs. (2) Treelike CP proofs and BoundeddepthFrege proofs cannot polynomially simulate each other. Our proofs also work for some generalizations of the CP proof system. In particular, they work for CP with a deduction rule, and also for proof systems that allow any formula with small communication complexity, and any set of sound rules of inference. 1 Introduction One of the most fundamental questions in pro...
Monotone Circuits for Matching Require Linear Depth
"... We prove that monotone circuits computing the perfect matching function on nvertex graphs require\Omega\Gamma n) depth. This implies an exponential gap between the depth of monotone and nonmonotone circuits. ..."
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Cited by 73 (8 self)
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We prove that monotone circuits computing the perfect matching function on nvertex graphs require\Omega\Gamma n) depth. This implies an exponential gap between the depth of monotone and nonmonotone circuits.