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Averagecase computational complexity theory
 Complexity Theory Retrospective II
, 1997
"... ABSTRACT Being NPcomplete has been widely interpreted as being computationally intractable. But NPcompleteness is a worstcase concept. Some NPcomplete problems are \easy on average", but some may not be. How is one to know whether an NPcomplete problem is \di cult on average"? The the ..."
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ABSTRACT Being NPcomplete has been widely interpreted as being computationally intractable. But NPcompleteness is a worstcase concept. Some NPcomplete problems are \easy on average", but some may not be. How is one to know whether an NPcomplete problem is \di cult on average"? The theory of averagecase computational complexity, initiated by Levin about ten years ago, is devoted to studying this problem. This paper is an attempt to provide an overview of the main ideas and results in this important new subarea of complexity theory. 1
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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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)....
The tale of oneway functions
 Problems of Information Transmission
, 2003
"... All the king’s horses, and all the king’s men, Couldn’t put Humpty together again. The existence of oneway functions (owf) is arguably the most important problem in computer theory. The article discusses and refines a number of concepts relevant to this problem. For instance, it gives the first com ..."
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All the king’s horses, and all the king’s men, Couldn’t put Humpty together again. The existence of oneway functions (owf) is arguably the most important problem in computer theory. The article discusses and refines a number of concepts relevant to this problem. For instance, it gives the first combinatorial complete owf, i.e., a function which is oneway if any function is. There are surprisingly many subtleties in basic definitions. Some of these subtleties are discussed or hinted at in the literature and some are overlooked. Here, a unified approach is attempted. 1
Randomizing Reductions Of Search Problems
 SIAM Journal of Computing
, 1993
"... . This paper closes a gap in the foundations of the theory of average case complexity. First, we clarify the notion of a feasible solution for a search problem and prove its robustness. Second, we give a general and usable notion of manyone randomizing reductions of search problems and prove that i ..."
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. This paper closes a gap in the foundations of the theory of average case complexity. First, we clarify the notion of a feasible solution for a search problem and prove its robustness. Second, we give a general and usable notion of manyone randomizing reductions of search problems and prove that it has desirable properties. All reductions of search problems to search problems in the literature on average case complexity can be viewed as such manyone randomizing reductions; this includes those reductions in the literature that use iterations and therefore do not look manyone. As an illustration, we present a careful proof in our framework of a theorem of Impagliazzo and Levin. Key words. Average case, search problems, reduction, randomization. 1. Introduction and results. Reduction theory for average case computational complexity was pioneered by Leonid Levin [?]. Recently, one of us wrote a survey on the subject [?], and we refer the reader there for a general background. However,...
Sets Computable in Polynomial Time on Average
, 1995
"... . In this paper, we discuss the complexity and properties of the sets which are computable in polynomialtime on average. This study is motivated by Levin's question of whether all sets in NP are solvable in polynomialtime on average for every reasonable (i.e., polynomialtime computable) distri ..."
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. In this paper, we discuss the complexity and properties of the sets which are computable in polynomialtime on average. This study is motivated by Levin's question of whether all sets in NP are solvable in polynomialtime on average for every reasonable (i.e., polynomialtime computable) distribution on the instances. Let PPcomp denote the class of all those sets which are computable in polynomialtime on average for every polynomialtime computable distribution on the instances. It is known that P ( PPcomp ( E. In this paper, we show that PPcomp is not contained in DTIME(2 cn ) for any constant c and that it lacks some basic structural properties: for example, it is not closed under manyone reducibility or for the existential operator. From these results, it follows that PPcomp contains Pimmune sets but no Pbiimmune sets; it is not included in P=cn for any constant c; and it is different from most of the wellknown complexity classes, such as UP, NP, BPP, and ...
Polynomial Time Samplable Distributions
 Proc. Mathematical Foundations of Computer Science
, 1995
"... This paper studies distributions which can be "approximated" by sampling algorithms in time polynomial in the length of their outputs. First, it is known that if polynomialtime samplable distributions are polynomialtime computable, then NP collapses to P. This paper shows by a simple counting argu ..."
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This paper studies distributions which can be "approximated" by sampling algorithms in time polynomial in the length of their outputs. First, it is known that if polynomialtime samplable distributions are polynomialtime computable, then NP collapses to P. This paper shows by a simple counting argument that every polynomialtime samplable distribution is computable in polynomial time if and only if so is every #P function. By this result, the class of polynomially samplable distributions contains no universal distributions if FP = #P. Second, it is also known that there exists polynomially samplable distributions which are not polynomially dominated by any polynomialtime computable distribution if strongly oneway functions exist. This paper strengthens this statement and shows that an assumption, namely, NP 6` BPP leads to the same consequence. Third, this paper shows that P = NP follows from the assumption that every polynomialtime samplable distribution is polynomially equivalent...
Truthtable closure and Turing closure of average polynomial time have different measures in EXP
 In Proceedings of the Eleventh Annual IEEE Conference on Computational Complexity
, 1996
"... Let PPcomp denote the sets that are solvable in polynomial time on average under every polynomialtime computable distribution on the instances. In this paper we show that the truthtable closure of PPcomp has measure 0 in EXP. Since, as we show, EXP is Turing reducible to PPcomp , the Turing clo ..."
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Let PPcomp denote the sets that are solvable in polynomial time on average under every polynomialtime computable distribution on the instances. In this paper we show that the truthtable closure of PPcomp has measure 0 in EXP. Since, as we show, EXP is Turing reducible to PPcomp , the Turing closure has measure 1 in EXP and thus, PPcomp is an example of a subclass of E such that the closure under truthtable reduction and the closure under Turing reduction have different measures in EXP. Furthermore, it is shown that there exists a set A in PPcomp such that for every k, the class of sets L such that A is ktruthtable reducible to L has measure 0 in EXP. 1 Introduction A randomized problem (or distributional problem) is a pair consisting of a decision problem and a density function. A randomized decision problem (A; ¯) is solvable in average polynomial time ((A; ¯) is in AP) if there exists a deterministic Turing machine M such that A = L(M ) and TimeM , the running time of M ...
The Complexity of Generating Test Instances
 in Proc. STACS'97, Lecture Notes in Computer Science
, 1997
"... Recently, Watanabe proposed a new framework for testing the correctness and average case behavior of algorithms that purport to solve a given NP search problem efficiently on average. The idea is to randomly generate certified instances in a way that resembles the underlying distribution ¯. We discu ..."
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Recently, Watanabe proposed a new framework for testing the correctness and average case behavior of algorithms that purport to solve a given NP search problem efficiently on average. The idea is to randomly generate certified instances in a way that resembles the underlying distribution ¯. We discuss this approach and show that test instances can be generated for every NP search problem with nonadaptive queries to an NP oracle. Further, we introduce Las Vegas as well as Monte Carlo types of test instance generators and show that these generators can be used to find out whether an algorithm is correct and efficient on average under ¯. In fact, it is not hard to construct Monte Carlo generators for all RP search problems as well as Las Vegas generators for all ZPP search problems. On the other hand, we prove that Monte Carlo generators can only exist for problems in NP " coAM. 1 Introduction The class NP plays a central role in computational complexity theory since it contains a larg...
LR(k) Testing is AverageCase Complete
"... In this note, we show that LR(k) testing of contextfree grammars with respect to random instances is manyone complete for DistNP (Distributional NP). The same result holds for testing whether a contextfree grammar is LL(k), strong LL(k), SLR(k), LC(k) or strong LC(k), respectively. 1 Introduction ..."
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In this note, we show that LR(k) testing of contextfree grammars with respect to random instances is manyone complete for DistNP (Distributional NP). The same result holds for testing whether a contextfree grammar is LL(k), strong LL(k), SLR(k), LC(k) or strong LC(k), respectively. 1 Introduction Since its introduction in the early 1970's, the theory of NPcompleteness is one of the most studied topics in computational complexity theory. A problem is NPcomplete if it captures somehow the whole complexity of the class NP in the sense that every efficient algorithm for this problem is an efficient algorithm for every problem in NP. Today, more than 20 years later, hundreds of NPcomplete problems from all areas of computer science and discrete mathematics are known. For none of them an efficient (namely, a deterministic polynomially timebounded) algorithm was found. It seems that NPcomplete problems are indeed difficult to solve. However, the question whether P = NP, one of the fa...