Proof of the Satisfiability Conjecture for Large k

Ding, Jian; Sly, Allan; Sun, Nike

doi:10.1145/2746539.2746619

Cited by 136 publications

(175 citation statements)

References 64 publications

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“…Another strong indication of the correctness of the physicists' picture is that this asymptotic behaviour was recently proved rigorously by Coja-Oghlan and Panagiotou in [25] and, soon after, the exact threshold for large enough K (which is a stronger non-asymptotic result) was proved by Ding, Sly and Sun in [36].…”

Section: Minimum Matchingmentioning

confidence: 80%

“…The sequence of these random weights (v α ) α∈N r is called the Ruelle Probability Cascades (RPC) corresponding to the parameters (37). The probabilistic part of the physicists' picture is saying exactly that the Gibbs weights G(H α ) of the pure states H α , which are q r -clusters indexed by α ∈ N r , are equal in distribution to the Ruelle Probability Cascades with the parameters defined in (36) in terms of the functional order parameter ζ. In other words, we can generate them as…”

Section: Structure Of Asymptotic Gibbs Distributionsmentioning

confidence: 99%

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Introduction to the SK model

Panchenko

2014

Current Developments in Mathematics

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Section: Minimum Matchingmentioning

confidence: 80%

Section: Structure Of Asymptotic Gibbs Distributionsmentioning

confidence: 99%

Introduction to the SK model

Panchenko

2014

Current Developments in Mathematics

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“…Theoretical investigations about the source and nature of this phase transition in the K-SAT problem have revolutionized our understanding from this problem and its state space geometry. Recently threshold conjecture has been proved analytically for some specific conditions [30].…”

Section:    mentioning

confidence: 99%

Relaxed Random Search for Solving K-Satisfiability and its Information Theoretic Interpretation

Nayyeri¹,

Dastghaibyfard²

2017

ijacsa

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Abstract-The problem of finding satisfying assignments for conjunctive normal formula with K literals in each clause, known as K-SAT, has attracted many attentions in the previous three decades. Since it is known as NP-Complete Problem, its effective solution (finding solution within polynomial time) would be of great interest due to its relation with the most well-known open problem in computer science (P=NP Conjecture). Different strategies have been developed to solve this problem but in all of them the complexity is preserved in NP class. In this paper, by considering the recent approach of applying statistical physic methods for analyzing the phase transition in the complexity of algorithms used for solving K-SAT, we try to compute the complexity of using randomized algorithm for finding the solution of K-SAT in more relaxed regions. It is shown how the probability of literal flipping process can change the complexity of algorithm substantially. An information theoretic interpretation of this reduction in time complexity will be argued.

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“…Even though the sharp transition framework presented in the next section is widely applicable in theoretical computer science and allows one to deduce that the transition width of certain functions goes to 0, establishing the existence of a threshold and determining its precise location if it exists can be notoriously difficult 4 [70]- [72].…”

Section: Capacity-achieving Codesmentioning

confidence: 99%

Reed-Muller codes achieve capacity on erasure channels

Kudekar

Kumar

Mondelli

et al. 2016

Proceedings of the Forty-Eighth Annual ACM Symposium on Theory of Computing

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Abstract-We introduce a new approach to proving that a sequence of deterministic linear codes achieves capacity on an erasure channel under maximum a posteriori decoding. Rather than relying on the precise structure of the codes our method exploits code symmetry. In particular, the technique applies to any sequence of linear codes where the blocklengths are strictly increasing, the code rates converge, and the permutation group of each code is doubly transitive. In other words, we show that symmetry alone implies near-optimal performance.An important consequence of this result is that a sequence of Reed-Muller codes with increasing blocklength and converging rate achieves capacity. This possibility has been suggested previously in the literature but it has only been proven for cases where the limiting code rate is 0 or 1. Moreover, these results extend naturally to all affine-invariant codes and, thus, to extended primitive narrow-sense BCH codes. This also resolves, in the affirmative, the existence question for capacity-achieving sequences of binary cyclic codes. The primary tools used in the proof are the sharp threshold property for symmetric monotone boolean functions and the area theorem for extrinsic information transfer functions.

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Proof of the Satisfiability Conjecture for Large k

Cited by 136 publications

References 64 publications

Introduction to the SK model

Introduction to the SK model

Relaxed Random Search for Solving K-Satisfiability and its Information Theoretic Interpretation

Reed-Muller codes achieve capacity on erasure channels

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