{"type":"ScholarlyArticle","title":"The Hard Problems Are Almost Everywhere For Random CNF-XOR Formulas","authors":[{"name":"Jeffrey M. Dudek","slug":"jeffrey-m-dudek","given":"Jeffrey M.","family":"Dudek","affiliation":"Rice University"},{"name":"Kuldeep S. Meel","slug":"kuldeep-s-meel","given":"Kuldeep S.","family":"Meel","affiliation":"Rice Univesity"},{"name":"Moshe Y. Vardi","slug":"moshe-y-vardi","given":"Moshe Y.","family":"Vardi","affiliation":"Rice University"}],"abstract":"Recent universal-hashing based approaches to sampling and counting crucially depend on the runtime performance of SAT solvers on formulas expressed as the conjunction of both CNF constraints and variable-width XOR constraints (known as CNF-XOR formulas). In this paper, we present the first study of the runtime behavior of SAT solvers equipped with XOR-reasoning techniques on random CNF-XOR formulas. We empirically demonstrate that a state-of-the-art SAT solver scales exponentially on random CNF-XOR formulas across a wide range of XOR-clause densities, peaking around the empirical phase-transition location. On the theoretical front, we prove that the solution space of a random CNF-XOR formula 'shatters' at all nonzero XOR-clause densities into well-separated components, similar to the behavior seen in random CNF formulas known to be difficult for many SAT algorithms.","doi":"10.24963/ijcai.2017/84","url":"https://ijcai.org/proceedings/2017/84","pdf":"https://ijcai.org/proceedings/2017/0084.pdf","markdown":"https://ijcai.org/proceedings/2017/84.md","bibtex":"https://ijcai.org/proceedings/2017/bibtex/84","volume":{"slug":"2017","title":"Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence","short_title":"IJCAI 2017","url":"https://ijcai.org/proceedings/2017/","year":2017,"location":"Melbourne, Australia","dates":"19-25 August 2017","editors":"Carles Sierra","isbn":"978-0-9992411-0-3","publisher":"International Joint Conferences on Artificial Intelligence Organization","doi_prefix":"10.24963/ijcai.2017"},"section":"Constraints and Satisfiability","track":"Main track","number":84,"pages":{"first":600,"last":606},"keywords":["Constraints and Satisfiability: Constraints and Satisfiability","Constraints and Satisfiability: Constraint Satisfaction"],"videos":[],"references":[{"position":1,"raw":"[Achlioptas and Menchaca-Mendez, 2012] Dimitris Achlioptas and Ricardo Menchaca-Mendez. 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