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13
On the Theory of Average Case Complexity
 Journal of Computer and System Sciences
, 1997
"... This paper takes the next step in developing the theory of average case complexity initiated by Leonid A Levin. Previous works [Levin 84, Gurevich 87, Venkatesan and Levin 88] have focused on the existence of complete problems. We widen the scope to other basic questions in computational complexity. ..."
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Cited by 106 (7 self)
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This paper takes the next step in developing the theory of average case complexity initiated by Leonid A Levin. Previous works [Levin 84, Gurevich 87, Venkatesan and Levin 88] have focused on the existence of complete problems. We widen the scope to other basic questions in computational complexity. Our results include: ffl the equivalence of search and decision problems in the context of average case complexity; ffl an initial analysis of the structure of distributionalNP (i.e. NP problems coupled with "simple distributions") under reductions which preserve average polynomialtime; ffl a proof that if all of distributionalNP is in average polynomialtime then nondeterministic exponentialtime equals deterministic exponential time (i.e., a collapse in the worst case hierarchy); ffl definitions and basic theorems regarding other complexity classes such as average logspace. An exposition of the basic definitions suggested by Levin and suggestions for some alternative definitions ...
Simple Strategies for Large ZeroSum Games with Applications to Complexity Theory
 STOC 94
, 1994
"... Von Neumann’s MinMax Theorem guarantees that each player of a zerosum matrix game hss an optimal mixed strategy. We show that each player has a nearoptimal mixed strategy that chooses uniformly from a multiset of pure strategies of size logarithmic in the number of pure strategies available to th ..."
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Cited by 23 (2 self)
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Von Neumann’s MinMax Theorem guarantees that each player of a zerosum matrix game hss an optimal mixed strategy. We show that each player has a nearoptimal mixed strategy that chooses uniformly from a multiset of pure strategies of size logarithmic in the number of pure strategies available to the opponent. Thus, for exponentially large games, for which even representing an optimal mixed strategy can require exponential space, there are nearoptimal, linearsize strategies. These strategies are eaay to play and serve as small witnesses to the approximate value of the game. Because of the fundamental role of games, we expect this theorem to have many applications in complexity theory and cryptography. We use it to strengthen the connection established by Yao between randomized and distributional complexity and to obtain the following results: (1) Every language has anticheckers — small hard multisets of inputs certifying that small circuits can’t decide the language. (2) Circuits of a given size can generate random instances that are hard for all circuits of linearly smaller size. (3) Given an oracle M for any exponentially large game, the approximate value of the game and nearoptimal strategies for it can be computed in I&‘(M). (4) For any NPcomplete language L, the problems of (a) computing a hard distribution of instances of L and (b) estimating the circuit complexity of L are both in Z;.
Matrix Transformation is Complete for the Average Case
 SIAM Journal on Computing
, 1995
"... In the theory of worst case complexity, NP completeness is used to establish that, for all practical purposes, the given NP problem is not decidable in polynomial time. In the theory of average case complexity, average case completeness is supposed to play the role of NP completeness. However, the a ..."
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Cited by 20 (1 self)
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In the theory of worst case complexity, NP completeness is used to establish that, for all practical purposes, the given NP problem is not decidable in polynomial time. In the theory of average case complexity, average case completeness is supposed to play the role of NP completeness. However, the average case reduction theory is still at an early stage, and only a few average case complete problems are known. We present the first algebraic problem complete for the average case under a natural probability distribution. The problem is this: Given a unimodular matrix X of integers, a set S of linear transformations of such unimodular matrices and a natural number n, decide if there is a product of n (not necessarily different) members of S that takes X to the identity matrix. 1 Introduction The theory of NP completeness is very useful. It allows one to establish that certain NP problems are NP complete and therefore, for all practical purposes, not decidable in polynomial time (PTime)....
Notes on Levin's Theory of AverageCase Complexity
 Electronic Colloquium on Computational Complexity
, 1997
"... Abstract. In 1984, Leonid Levin initiated a theory of averagecase complexity. We provide an exposition of the basic definitions suggested by Levin, and discuss some of the considerations underlying these definitions. Keywords: Averagecase complexity, reductions. This survey is rooted in the author ..."
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Cited by 18 (2 self)
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Abstract. In 1984, Leonid Levin initiated a theory of averagecase complexity. We provide an exposition of the basic definitions suggested by Levin, and discuss some of the considerations underlying these definitions. Keywords: Averagecase complexity, reductions. This survey is rooted in the author’s (exposition and exploration) work [4], which was partially reproduded in [1]. An early version of this survey appeared as TR97058 of ECCC. Some of the perspective and conclusions were revised in light of a relatively recent work of Livne [21], but an attempt was made to preserve the spirit of the original survey. The author’s current perspective is better reflected in [7, Sec. 10.2] and [8], which advocate somewhat different definitional choices (e.g., focusing on typical rather than average performace of algorithms). 1
Bounded Depth Arithmetic Circuits: Counting and Closure
, 1999
"... Constantdepth arithmetic circuits have been defined and studied in [AAD97, ABL98]; these circuits yield the function classes #AC . These function classes in turn provide new characterizations of the computational power of threshold circuits, and provide a link between the circuit classes AC ..."
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Cited by 9 (3 self)
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Constantdepth arithmetic circuits have been defined and studied in [AAD97, ABL98]; these circuits yield the function classes #AC . These function classes in turn provide new characterizations of the computational power of threshold circuits, and provide a link between the circuit classes AC (where many lower bounds are known) and TC (where essentially no lower bounds are known). In this paper, we resolve several questions regarding the closure properties of #AC .
All Natural NPC Problems Have AverageCase Complete Versions
 IN 35TH ACM SYMPOSIUM ON THE THEORY OF COMPUTING
, 2006
"... In 1984 Levin put forward a suggestion for a theory of average case complexity. In this theory a problem, called a distributional problem, is defined as a pair consisting of a decision problem and a probability distribution over the instances. Introducing adequate notions of ”efficiencyonaverage”, ..."
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Cited by 4 (0 self)
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In 1984 Levin put forward a suggestion for a theory of average case complexity. In this theory a problem, called a distributional problem, is defined as a pair consisting of a decision problem and a probability distribution over the instances. Introducing adequate notions of ”efficiencyonaverage”, simple distributions and efficiencyonaverage preserving reductions, Levin developed a theory analogous to the theory of N Pcompleteness. In particular, he showed that there exists a simple distributional problem that is complete under these reductions. But since then very few distributional problems were shown to be complete in this sense. In this paper we show a simple sufficient condition for an N Pcomplete decision problem to have a distributional version that is complete under these reductions (and thus to be ”hard on the average ” with respect to some simple probability distribution). Apparently all known N Pcomplete decision problems meet this condition.
Efficient AverageCase Algorithms for the Modular Group
 In the Proceedings of The 35th Annual Symposium on Foundations of Computer Science
, 1994
"... The modular group occupies a central position in many branches of mathematical sciences. In this paper we give average polynomialtime algorithms for the unbounded and bounded membership problems for finitely generated subgroups of the modular group. The latter result affirms a conjecture of Gurevic ..."
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Cited by 4 (1 self)
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The modular group occupies a central position in many branches of mathematical sciences. In this paper we give average polynomialtime algorithms for the unbounded and bounded membership problems for finitely generated subgroups of the modular group. The latter result affirms a conjecture of Gurevich [5]. 1 Introduction 1.1 The Modular Group The modular group \Gamma is a remarkable mathematical object. It has several equivalent characterizations: (i) SL 2 ()= \Sigma I, the quotient of the group SL 2 () of 2 \Theta 2 integer matrices with determinant 1 modulo its central subgroup f\SigmaI g; (ii) the group of complex fractional linear transformations z 7! az + b cz + d with integer coefficients satisfying ad \Gamma bc = 1; (iii) the free product of cyclic groups of order 2 and 3; i.e., the group presented by generators R; S and relations R 2 j S 3 j 1; (iv) the group of automorphisms of a certain regular tesselation of the hyperbolic plane (Figure 1); Proc. 35th IEEE Symp...
AverageCase Complexity Theory and PolynomialTime Reductions
, 2001
"... This thesis studies averagecase complexity theory and polynomialtime reducibilities. The issues in averagecase complexity arise primarily from Cai and Selman's extension of Levin's denition of average polynomial time. We study polynomialtime reductions between distributional problems. Under stro ..."
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Cited by 2 (0 self)
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This thesis studies averagecase complexity theory and polynomialtime reducibilities. The issues in averagecase complexity arise primarily from Cai and Selman's extension of Levin's denition of average polynomial time. We study polynomialtime reductions between distributional problems. Under strong but reasonable hypotheses we separate ordinary NPcompleteness notions.
The Bounded Membership Problem of the Monoid SL_2(N)
"... SL_2(N ) is the set of all 2 \Theta 2 matrices with nonnegative integer entries and determinant 1. The bounded membership problem BMN of SL 2 (N ) is that given a subset S of SL 2 (N ), a matrix A 2 SL 2 (N ), and an integer n, whether A can be represented as a product of at most n matrices (repeti ..."
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Cited by 1 (0 self)
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SL_2(N ) is the set of all 2 \Theta 2 matrices with nonnegative integer entries and determinant 1. The bounded membership problem BMN of SL 2 (N ) is that given a subset S of SL 2 (N ), a matrix A 2 SL 2 (N ), and an integer n, whether A can be represented as a product of at most n matrices (repetitions are allowed) in S. We prove that BMN is NPcomplete, but its randomized version under natural distribution is solvable in average polynomial time. Furthermore, it is proved that if the number of elements of S is a constant, then BMN is polynomial time computable.
Membership Problem for the Modular Group
, 2007
"... The modular group plays an important role in many branches of mathematics. We show that the membership problem for the modular group is decidable in polynomial time. To this end, we develop a new syllablebased version of the known subgroupgraph approach. The new approach can be used to prove addi ..."
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Cited by 1 (0 self)
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The modular group plays an important role in many branches of mathematics. We show that the membership problem for the modular group is decidable in polynomial time. To this end, we develop a new syllablebased version of the known subgroupgraph approach. The new approach can be used to prove additional results. We demonstrate this by using it to prove that the membership problem for a free group remains decidable in polynomial time when elements are written in a normal form with exponents.