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Strict Polynomialtime in Simulation and Extraction
, 2004
"... The notion of efficient computation is usually identified in cryptography and complexity with (strict) probabilistic polynomial time. However, until recently, in order to obtain constantround ..."
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Cited by 43 (8 self)
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The notion of efficient computation is usually identified in cryptography and complexity with (strict) probabilistic polynomial time. However, until recently, in order to obtain constantround
On approximating optimal weighted lobbying, and frequency of correctness versus averagecase polynomial time
, 2007
"... We investigate issues related to two hard problems related to voting, the optimal weighted lobbying problem and the winner problem for Dodgson elections. Regarding the former, Christian et al. [CFRS06] showed that optimal lobbying is intractable in the sense of parameterized complexity. We provide a ..."
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Cited by 14 (6 self)
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We investigate issues related to two hard problems related to voting, the optimal weighted lobbying problem and the winner problem for Dodgson elections. Regarding the former, Christian et al. [CFRS06] showed that optimal lobbying is intractable in the sense of parameterized complexity. We provide an efficient greedy algorithm that achieves a logarithmic approximation ratio for this problem and even for a more general variant—optimal weighted lobbying. We prove that essentially no better approximation ratio than ours can be proven for this greedy algorithm. The problem of determining Dodgson winners is known to be complete for parallel access to NP [HHR97]. Homan and Hemaspaandra [HH06] proposed an efficient greedy heuristic for finding Dodgson winners with a guaranteed frequency of success, and their heuristic is a “frequently selfknowingly correct algorithm. ” We prove that every distributional problem solvable in polynomial time on the average with respect to the uniform distribution has a frequently selfknowingly correct polynomialtime algorithm. Furthermore, we study some features of probability weight of correctness with respect to Procaccia and Rosenschein’s junta distributions [PR07]. Key words: approximation, Dodgson elections, election systems, frequently selfknowingly correct algorithms, greedy algorithms, optimal lobbying, preference aggregation.
Computational Complexity
, 2004
"... The strive for efficiency is ancient and universal, as time is always short for humans. Computational Complexity is a mathematical study of the what can be achieved when time (and other resources) are scarce. In this ..."
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Cited by 9 (1 self)
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The strive for efficiency is ancient and universal, as time is always short for humans. Computational Complexity is a mathematical study of the what can be achieved when time (and other resources) are scarce. In this
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.
NP and Mathematics  a computational complexity perspective
 Proc. of the ICM 06
"... “P versus N P – a gift to mathematics from Computer Science” ..."
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“P versus N P – a gift to mathematics from Computer Science”
Frequency of Correctness versus Average Polynomial Time and Generalized Juntas ∗
, 2008
"... We prove that every distributional problem solvable in polynomial time on the average with respect to the uniform distribution has a frequently selfknowingly correct polynomialtime algorithm. We also study some features of probability weight of correctness with respect to generalizations of Procac ..."
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Cited by 1 (1 self)
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We prove that every distributional problem solvable in polynomial time on the average with respect to the uniform distribution has a frequently selfknowingly correct polynomialtime algorithm. We also study some features of probability weight of correctness with respect to generalizations of Procaccia and Rosenschein’s junta distributions [PR07b]. Key words: frequently selfknowingly correct algorithms, greedy algorithms, junta distributions.
A Theory of AverageCase Compilability in Knowledge Representation
"... Compilability is a fundamental property of knowledge representation formalisms which captures how succinctly information can be expressed. Although many results concerning compilability have been obtained, they are all "worstcase" results. ..."
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Compilability is a fundamental property of knowledge representation formalisms which captures how succinctly information can be expressed. Although many results concerning compilability have been obtained, they are all "worstcase" results.
Average Case Complexity, Revisited
"... Abstract. More than two decades elapsed since Levin set forth a theory of averagecase complexity. In this survey we present the basic aspects of this theory as well as some of the main results regarding it. The current presentation deviates from our old “Notes on Levin’s Theory of AverageCase Comp ..."
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Abstract. More than two decades elapsed since Levin set forth a theory of averagecase complexity. In this survey we present the basic aspects of this theory as well as some of the main results regarding it. The current presentation deviates from our old “Notes on Levin’s Theory of AverageCase Complexity ” (ECCC, TR97058, 1997) in several aspects. In particular: – We currently view averagecase complexity as referring to the performance on “average ” (or rather typical) instances, and not as the average performance on random instances. (Thus, it may be more justified to refer to this theory by the name typicalcase complexity, but we retain the name averagecase for historical reasons.) – We include a treatment of search problems, and a presentation of the reduction of “NP with sampleable distributions ” to “NP with Pcomputable distributions ” (due to Impagliazzo and Levin, 31st FOCS, 1990). – We include Livne’s result (ECCC, TR06122, 2006) by which all natural NPCproblems have averagecase complete versions. This result seems to shed doubt on the association of Pcomputable distributions with natural distributions. Keywords: AverageCase Complexity. This text has been revised based on [6, Sec. 10.2].
Introduction to Computational Complexity ∗ — Mathematical Programming Glossary Supplement —
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P ≠ RP Proof
"... Abstract. This paper demonstrates that P ≠ RP. The way was to generalize the traditional definitions of the classes P and RP, to construct an artificial problem (Maj2/3XGSAT) and then to demonstrate that it is in RP but not in P (where the classes P and RP are generalized and called too simply P a ..."
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Abstract. This paper demonstrates that P ≠ RP. The way was to generalize the traditional definitions of the classes P and RP, to construct an artificial problem (Maj2/3XGSAT) and then to demonstrate that it is in RP but not in P (where the classes P and RP are generalized and called too simply P and RP in this paper, and then it is explained why the traditional classes P and RP should be fixed and replaced by these generalized ones into Theory of Computer Science). The demonstration consists of: 1. Definition of Restricted Type X Program 2. Definition of the Majority2/3 Extended General Satisfiability – Maj2/3XGSAT 3. Generalization to classes P and RP 4. Demonstration that the Maj2/3XGSAT is in RP 5. Demonstration that the Maj2/3XGSAT is not in P 6. Demonstration that the BakerGillSolovay Theorem does not refute the proof 7. Demonstration that the RazborovRudich Theorem does not refute the proof