{"type":"ScholarlyArticle","title":"On Division Versus Saturation in Pseudo-Boolean Solving","authors":[{"name":"Stephan Gocht","slug":"stephan-gocht","given":"Stephan","family":"Gocht","affiliation":"KTH Royal Institute of Technology"},{"name":"Jakob Nordström","slug":"jakob-nordstrom","given":"Jakob","family":"Nordström","affiliation":"University of Copenhagen; KTH Royal Institute of Technology"},{"name":"Amir Yehudayoff","slug":"amir-yehudayoff","given":"Amir","family":"Yehudayoff","affiliation":"Technion -Israel Institute of Technology"}],"abstract":"The conflict-driven clause learning (CDCL) paradigm has revolutionized\r\nSAT solving over the last two decades. Extending this approach to\r\npseudo-Boolean (PB) solvers doing 0-1 linear programming holds the\r\npromise of further exponential improvements in theory, but\r\nintriguingly such gains have not materialized in practice. Also\r\nintriguingly, most PB extensions of CDCL use not the division rule in\r\ncutting planes as defined in [Cook et al., '87] but instead the\r\nso-called saturation rule. To the best of our knowledge, there has\r\nbeen no study comparing the strengths of division and saturation in\r\nthe context of conflict-driven PB learning, when all linear\r\ncombinations of inequalities are required to cancel variables.\r\nWe show that PB solvers with division instead of saturation can be\r\nexponentially stronger. In the other direction, we prove that\r\nsimulating a single saturation step can require an exponential number\r\nof divisions. We also perform some experiments to see whether these\r\nphenomena can be observed in actual solvers. Our conclusion is that a\r\ncareful combination of division and saturation seems to be crucial to\r\nharness more of the power of cutting planes.","doi":"10.24963/ijcai.2019/237","url":"https://ijcai.org/proceedings/2019/237","pdf":"https://ijcai.org/proceedings/2019/0237.pdf","markdown":"https://ijcai.org/proceedings/2019/237.md","bibtex":"https://ijcai.org/proceedings/2019/bibtex/237","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":"Knowledge Representation and Reasoning","track":"Main track","number":237,"pages":{"first":1711,"last":1718},"keywords":["Constraints and SAT: SAT: Evaluation and Analysis","Constraints and SAT: Constraints: Evaluation and Analysis","Heuristic Search and Game Playing: Combinatorial Search and Optimisation","Knowledge Representation and Reasoning: Computational Complexity of Reasoning"],"videos":[],"references":[{"position":1,"raw":"[Atserias et al., 2011] Albert Atserias, Johannes Klaus Fichte, and Marc Thurley. 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