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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"? ..."
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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&quot;, but some may not be. How is one to know whether an NPcomplete problem is \di cult on average&quot;? 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.
On Average Case Complexity of SAT for Symmetric Distributions
, 1995
"... We investigate in this paper 'natural' distributions for the satisfiability problem (SAT) of propositional logic, using concepts introduced by [25, 19, 1] to study the averagecase complexity of NPcomplete problems. Gurevich showed that a problem with a flat distribution is not DistNP com ..."
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We investigate in this paper 'natural' distributions for the satisfiability problem (SAT) of propositional logic, using concepts introduced by [25, 19, 1] to study the averagecase complexity of NPcomplete problems. Gurevich showed that a problem with a flat distribution is not DistNP complete (for deterministic reductions), unless DEXPTime<F NaN> 6= NEXPTime. We express the known results concerning fixed size and fixed density distributions for CNF in the framework of averagecase complexity and show that all these distributions are flat. We introduce the family of symmetric distributions, which generalizes those mentioned before, and show that bounded symmetric distributions on ordered tuples of clauses (CNFTuples) and on kCNF (sets of kliteralclauses), are flat. This eliminates all these distributions as candidates for 'provably hard' (i.e. DistNP complete) distributions for SAT, if one considers only deterministic reductions. Given the (presumed) naturalness and generality o...
Some Results on Derandomization
 Theory of Computing Systems
"... We show several results about derandomization including 1. If NP is easy on average then e#cient pseudorandom generators exist and P = BPP. ..."
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We show several results about derandomization including 1. If NP is easy on average then e#cient pseudorandom generators exist and P = BPP.
On the Complexity of Deadlock Detection in Families of Planar Nets
, 1995
"... We are interested in some properties of massively parallel computers that we model by finite automata connected together as a 2dimensional grid. We wonder whether it is possible to anticipate a possible appearance of a deadlock in such nets. Thus, we look for efficient algorithms to predict whe ..."
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We are interested in some properties of massively parallel computers that we model by finite automata connected together as a 2dimensional grid. We wonder whether it is possible to anticipate a possible appearance of a deadlock in such nets. Thus, we look for efficient algorithms to predict whether deadlocks can appear in grids of bounded size. From the point of view of worstcase complexity, we prove that this problem is NPcomplete whereas it is quadratic for linear structures. The method we use is a reduction from a tiling problem. We also prove that this problem, associated with a natural probability distribution on its instances, is RNPcomplete (Random NPcomplete) in the theory proposed by Levin and Gurevich. Very few randomized problems are known to be RNPcomplete. Under classical complexity hypotheses, this result proves that there does not exist any algorithm that solves this problem efficiently on average case. We present others extentions of our results for d...
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.
Wang Randomized Reductions (Info) The Chicago Journal of Theoretical Computer Science is abstracted or indexed
, 1999
"... Published one article at a time in L ATEX source form on the ..."
Karg, Köbler, and Schuler Complexity of Generating Test Instances (Info) The Chicago Journal of Theoretical Computer Science is abstracted or indexed
, 1999
"... Published one article at a time in L ATEX source form on the Internet. Pagination ..."
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Published one article at a time in L ATEX source form on the Internet. Pagination
The Average Case Complexity of Multilevel Syllogistic
"... An approach to the problem of developing provably correct programs has been to enrich a theorem prover for Hoare logic with decision procedures for a number of decidable sublanguages of set theory ( EMLS, MLS, and extensions) and arithmetic (FPILP ) [Sch77]. Citing results of Goldberg [Gol79] on ..."
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An approach to the problem of developing provably correct programs has been to enrich a theorem prover for Hoare logic with decision procedures for a number of decidable sublanguages of set theory ( EMLS, MLS, and extensions) and arithmetic (FPILP ) [Sch77]. Citing results of Goldberg [Gol79] on average case behavior of algorithms for SAT , it was hoped that these decision procedures would perform well on average. So far, it has been fairly difficult to prove average case NPhardness under the various definitions ([Lev86], [BDCGL89], [BG91], [Gur91], [VR92], [SY92], [RS93]). We should note that the definitions in the literature haven't yet been standardized. We survey some of the results of the average case analysis of NPcomplete problems, and compare the results of Goldberg with more pessimistic results. We prove that FPILP , EMLS and related fragments of set theory are NPaverage complete, and show that there are simple distributions that will frustrate any algorithm fo...
Randomized Reductions and Isomorphisms
, 1999
"... Randomizing reductions have provided new techniques for tackling averagecase complexity problems. For example, although some NPcomplete problems with uniform distributions on instances cannot be complete under deterministic oneone reductions [WB95], they are complete under randomized reductions [ ..."
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Randomizing reductions have provided new techniques for tackling averagecase complexity problems. For example, although some NPcomplete problems with uniform distributions on instances cannot be complete under deterministic oneone reductions [WB95], they are complete under randomized reductions [VL88]. We study randomized reductions in this paper on reductions that are oneone and honest mappings over certain input domains. These are reasonable assumptions since all the randomized reductions in the literature that are used in proving averagecase completeness results possess this property. We consider whether randomized reductions can be inverted efficiently. This gives rise to the issue of randomized isomorphisms. By generalizing the notion of isomorphisms under deterministic reductions, we define what it means for two distributional problems to be isomorphic under randomized reductions. We then show a randomized version of the CantorBernsteinMyhill theorem, which pro...