{"type":"ScholarlyArticle","title":"Sharpness of the Satisfiability Threshold for Non-Uniform Random k-SAT","authors":[{"name":"Tobias Friedrich","slug":"tobias-friedrich","given":"Tobias","family":"Friedrich","affiliation":"Hasso Plattner Institute , University of Potsdam , Potsdam , Germany"},{"name":"Ralf Rothenberger","slug":"ralf-rothenberger","given":"Ralf","family":"Rothenberger","affiliation":"Hasso Plattner Institute , University of Potsdam , Potsdam , Germany"}],"abstract":"We study a more general model to generate random instances of Propositional Satisfiability (SAT) with n Boolean variables, m clauses, and exactly k variables per clause. Additionally, our model is given an arbitrary probability distribution (p_1, ..., p_n) on the variable occurrences. Therefore, we call it non-uniform random k-SAT. The number m of randomly drawn clauses at which random formulas go from asymptotically almost surely (a.a.s.) satisfiable to a.a.s. unsatisfiable is called the satisfiability threshold. Such a threshold is called sharp if it approaches a step function as n increases. \r\n\r\nWe identify conditions on the variable probability distribution (p_1, ..., p_n) under which the satisfiability threshold is sharp if its position is already known asymptotically. This result generalizes Friedgut’s sharpness result from uniform to non-uniform random k -SAT and implies sharpness for thresholds of a wide range of random k -SAT models with heterogeneous probability distributions, for example such models where the variable probabilities follow a power-law.","doi":"10.24963/ijcai.2019/853","url":"https://ijcai.org/proceedings/2019/853","pdf":"https://ijcai.org/proceedings/2019/0853.pdf","markdown":"https://ijcai.org/proceedings/2019/853.md","bibtex":"https://ijcai.org/proceedings/2019/bibtex/853","volume":{"slug":"2019","title":"Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence","short_title":"IJCAI 2019","url":"https://ijcai.org/proceedings/2019/","year":2019,"location":"Macao","dates":"10-16 August 2019","editors":"Sarit Kraus","isbn":"978-0-9992411-4-1","publisher":"International Joint Conferences on Artificial Intelligence Organization","doi_prefix":"10.24963/ijcai.2019"},"section":"Best Sister Conferences","track":"Best Sister Conferences","number":853,"pages":{"first":6151,"last":6155},"keywords":["Constraints and SAT: SAT: Evaluation and Analysis","Constraints and SAT: Modeling;Formulation"],"videos":[],"references":[{"position":1,"raw":"[Ansótegui et al., 2009a] Carlos Ansótegui, Maria Luisa Bonet, and Jordi Levy. 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