Results 1  10
of
119
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 2347 (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...
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 127 (4 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.
Geometric Complexity Theory I: An Approach to the P. vs. NP and related problems
, 2001
"... We suggest an approach based on geometric invariant theory to the fundamentallower bound problems in complexity theory concerning formula and circuit size. Specifically, ..."
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Cited by 44 (12 self)
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We suggest an approach based on geometric invariant theory to the fundamentallower bound problems in complexity theory concerning formula and circuit size. Specifically,
Evaluating Signs of Determinants Using SinglePrecision Arithmetic
, 1994
"... We propose a method to evaluate signs of 2 x 2 and 3 x 3 determinants with bbit integer entries using only b and (b + 1)bit arithmetic respectively. This algorithm has numerous applications in geometric computation and provides a general and practical approach to robustness. The algorithm has been ..."
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Cited by 40 (5 self)
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We propose a method to evaluate signs of 2 x 2 and 3 x 3 determinants with bbit integer entries using only b and (b + 1)bit arithmetic respectively. This algorithm has numerous applications in geometric computation and provides a general and practical approach to robustness. The algorithm has been implemented and experimental results show that it slows down the computing time by only a small factor with respect to floatingpoint calculation.
NonCommutative Arithmetic Circuits: Depth Reduction and Size Lower Bounds
 Theoretical Computer Science
"... We investigate the phenomenon of depthreduction in commutativeand noncommutative arithmetic circuits. We prove that in the commutative setting, uniform semiunbounded arithmetic circuits of logarithmic depth are as powerful as uniform arithmetic circuits of polynomial degree (and unrestricted dept ..."
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Cited by 28 (11 self)
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We investigate the phenomenon of depthreduction in commutativeand noncommutative arithmetic circuits. We prove that in the commutative setting, uniform semiunbounded arithmetic circuits of logarithmic depth are as powerful as uniform arithmetic circuits of polynomial degree (and unrestricted depth); earlier proofs did not work in the uniform setting. This also provides a unified proof of the circuit characterizations of the class LOGCFL and its counting variant #LOGCFL. We show that AC 1 has no more power than arithmetic circuits of polynomial size and degree n O(log log n) (improving the trivial bound of n O(logn) ). Connections are drawn between TC 1 and arithmetic circuits of polynomial size and degree. Then we consider noncommutative computation. We show that over the algebra (\Sigma ; max, concat), arithmetic circuits of polynomial size and polynomial degree can be reduced to O(log 2 n) depth (and even to O(log n) depth if unboundedfanin gates are allowed) . This...
The Computational Complexity of Some Problems of Linear Algebra
 STACS '97
, 1997
"... We consider the computational complexity of some problems dealing with matrix rank. Let E; S be subsets of a commutative ring R. Let x 1 ; x 2 ; : : : ; x t be variables. Given a matrix M = M(x 1 ; x 2 ; : : : ; x t ) with entries chosen from E [ fx 1 ; x 2 ; : : : ; x t g, we want to determine ..."
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Cited by 24 (2 self)
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We consider the computational complexity of some problems dealing with matrix rank. Let E; S be subsets of a commutative ring R. Let x 1 ; x 2 ; : : : ; x t be variables. Given a matrix M = M(x 1 ; x 2 ; : : : ; x t ) with entries chosen from E [ fx 1 ; x 2 ; : : : ; x t g, we want to determine maxrank S (M) = max (a 1 ;a 2 ;:::;a t )2S t rank M(a 1 ; a 2 ; : : : a t ) and minrank S (M) = min (a 1 ;a 2 ;:::;a t )2S t rank M(a 1 ; a 2 ; : : : a t ): There are also variants of these problems that specify more about the structure of M , or instead of asking for the minimum or maximum
Counting complexity classes for numeric computations II: Algebraic and semialgebraic sets (Extended Abstract)
 J. COMPL
, 2004
"... We define counting #P classes #P ¡ and in the BlumShubSmale setting of computations over the real or complex numbers, respectively. The problems of counting the number of solutions of systems of polynomial inequalities over ¢ , or of systems of polynomial equalities over £ , respectively, turn ou ..."
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Cited by 17 (10 self)
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We define counting #P classes #P ¡ and in the BlumShubSmale setting of computations over the real or complex numbers, respectively. The problems of counting the number of solutions of systems of polynomial inequalities over ¢ , or of systems of polynomial equalities over £ , respectively, turn out to be natural complete problems in these classes. We investigate to what extent the new counting classes capture the complexity of computing basic topological invariants of semialgebraic sets (over ¢ ) and algebraic sets (over £). We prove that the problem to compute the (modified) Euler characteristic of semialgebraic sets is FP #P¤complete, and that the problem to compute the geometric degree of complex algebraic sets is FP #P¥complete. We also define new counting complexity classes GCR and GCC in the classical Turing model via taking Boolean parts of the classes above, and show that the problems to compute the Euler characteristic and the geometric degree of (semi)algebraic sets given by integer polynomials are complete in these classes. We complement the results in the Turing model by proving, for all k ¦ ∈ , the FPSPACEhardness of the problem of computing the kth Betti number of the set of real zeros of a given integer polynomial. This holds with respect to the singular homology as well as for the BorelMoore homology.
AN OVERVIEW OF MATHEMATICAL ISSUES ARISING IN THE GEOMETRIC COMPLEXITY THEORY APPROACH TO VP != VNP
"... We discuss the geometry of orbit closures and the asymptotic behavior of Kronecker coefficients in the context of the Geometric Complexity Theory program to prove a variant of Valiant’s algebraic analog of the P ̸ = NP conjecture. We also describe the precise separation of complexity classes that t ..."
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Cited by 15 (6 self)
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We discuss the geometry of orbit closures and the asymptotic behavior of Kronecker coefficients in the context of the Geometric Complexity Theory program to prove a variant of Valiant’s algebraic analog of the P ̸ = NP conjecture. We also describe the precise separation of complexity classes that their program proposes to demonstrate.
A Lower Bound for the Size of Syntactically Multilinear Arithmetic Circuits
 Proceedings of the 48th FOCS: 438–448
, 2007
"... We construct an explicit polynomial f(x1,...,xn), with coefficients in {0, 1}, such that the size of any syntactically multilinear arithmetic circuit computing f is at least Ω(n 4/3 / log 2 n). The lower bound holds over any field. ..."
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Cited by 14 (10 self)
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We construct an explicit polynomial f(x1,...,xn), with coefficients in {0, 1}, such that the size of any syntactically multilinear arithmetic circuit computing f is at least Ω(n 4/3 / log 2 n). The lower bound holds over any field.