{"type":"ScholarlyArticle","title":"Divide and Conquer: Towards Faster Pseudo-Boolean Solving","authors":[{"name":"Jan Elffers","slug":"jan-elffers","given":"Jan","family":"Elffers","affiliation":"KTH Royal Institute of Technology"},{"name":"Jakob Nordström","slug":"jakob-nordstrom","given":"Jakob","family":"Nordström","affiliation":"KTH Royal Institute of Technology"}],"abstract":"The last 20 years have seen dramatic improvements in the performance\r\nof algorithms for Boolean satisfiability---so-called SAT solvers---and\r\ntoday conflict-driven clause learning (CDCL) solvers are routinely\r\nused in a wide range of application areas.  One serious short-coming\r\nof CDCL, however, is that the underlying method of reasoning is quite\r\nweak.  A tantalizing solution is to instead use stronger\r\npseudo-Boolean (PB) reasoning, but so far the promise of exponential\r\ngains in performance has failed to materialize---the increased\r\ntheoretical strength seems hard to harness algorithmically, and in\r\nmany applications CDCL-based methods are still superior.  We propose a\r\nmodified approach to pseudo-Boolean solving based on division instead\r\nof the saturation rule used in [Chai and Kuehlmann '05] and other PB\r\nsolvers.  In addition to resulting in a stronger conflict analysis,\r\nthis also improves performance by keeping integer coefficient sizes\r\ndown, and yields a very competitive solver as shown by the results in\r\nthe Pseudo-Boolean Competitions 2015 and 2016.","doi":"10.24963/ijcai.2018/180","url":"https://ijcai.org/proceedings/2018/180","pdf":"https://ijcai.org/proceedings/2018/0180.pdf","markdown":"https://ijcai.org/proceedings/2018/180.md","bibtex":"https://ijcai.org/proceedings/2018/bibtex/180","volume":{"slug":"2018","title":"Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence","short_title":"IJCAI 2018","url":"https://ijcai.org/proceedings/2018/","year":2018,"location":"Stockholm","dates":"13-19 July 2018","editors":"Jérôme Lang","isbn":"978-0-9992411-2-7","publisher":"International Joint Conferences on Artificial Intelligence Organization","doi_prefix":"10.24963/ijcai.2018"},"section":"Constraints and SAT","track":"Main track","number":180,"pages":{"first":1291,"last":1299},"keywords":["Constraints and SAT: SAT","Constraints and SAT: Constraints and SAT","Constraints and SAT: SAT: Evaluation and Analysis","Constraints and SAT: SAT: Solvers and Tools"],"videos":[],"references":[{"position":1,"raw":"[Bayardo Jr. and Schrag, 1997] R. J. Bayardo Jr. and R. Schrag. Using CSP look-back techniques to solve real-world SAT in- stances. In Proc. 14th National Conf. Artif. Intell. (AAAI '97), pp. 203-208, 1997.","title":"Using CSP look-back techniques to solve real-world SAT instances","authors":"R J Bayardo, R Schrag","year":"1997","venue":"Proc. 14th National Conf. Artif. Intell. (AAAI '97)","doi":null,"url":null,"internal_url":null},{"position":2,"raw":"[Biere et al., 2009] A. Biere, M. J. H. Heule, H. van Maaren, and T. Walsh, editors. Handbook of Satisfiability. 2009.","title":"Handbook of Satisfiability","authors":null,"year":"2009","venue":null,"doi":null,"url":null,"internal_url":null},{"position":3,"raw":"[Biere et al., 2014] A. Biere, D. Le Berre, E. Lonca, and N. Man- they. Detecting cardinality constraints in CNF. In Proc. 17th Conf. Theory Appl. Sat. Testing (SAT '14), pp. 285-301, 2014.","title":"Detecting cardinality constraints in CNF","authors":"A Biere, D Le Berre, E Lonca, N Manthey","year":"2014","venue":"Proc. 17th Conf. Theory Appl. Sat. Testing (SAT '14)","doi":null,"url":null,"internal_url":null},{"position":4,"raw":"[Chai and Kuehlmann, 2005] D. Chai and A. Kuehlmann. A fast pseudo-Boolean constraint solver. IEEE Trans. Computer-Aided Design of Integrated Circuits and Systems, 24(3):305-317, 2005.","title":"A fast pseudo-Boolean constraint solver","authors":"D Chai, A Kuehlmann","year":"2005","venue":"IEEE Trans. Computer-Aided Design of Integrated Circuits and Systems","doi":null,"url":null,"internal_url":null},{"position":5,"raw":"[Cook et al., 1987] W. Cook, C. R. Coullard, and G. Turán. On the complexity of cutting-plane proofs. Disc. Appl. Math., 18(1):25-38, 1987.","title":"On the complexity of cutting-plane proofs","authors":"W Cook, C R Coullard, G Turán","year":"1987","venue":"Disc. Appl. Math","doi":null,"url":null,"internal_url":null},{"position":6,"raw":"Cook, 1971","title":null,"authors":"Cook","year":"1971","venue":null,"doi":null,"url":null,"internal_url":null},{"position":7,"raw":"S. A. Cook. The complexity of theorem-proving pro- cedures. In Proc. 3rd Symp. Theory of Computing (STOC '71), pp. 151-158, 1971.","title":"The complexity of theorem-proving procedures","authors":"S A Cook","year":"1971","venue":"Proc. 3rd Symp. Theory of Computing (STOC '71)","doi":null,"url":null,"internal_url":null},{"position":8,"raw":"[Dixon and Ginsberg, 2002] H. E. Dixon and M. L. Ginsberg. In- ference methods for a pseudo-Boolean satisfiability solver. In Proc. 18th National Conf. Artif. Intell. (AAAI '02), pp. 635-640, 2002.","title":"Inference methods for a pseudo-Boolean satisfiability solver","authors":"H E Dixon, M L Ginsberg","year":"2002","venue":"Proc. 18th National Conf. Artif. Intell. (AAAI '02)","doi":null,"url":null,"internal_url":null},{"position":9,"raw":"[Dixon et al., 2004] H. E. Dixon, M. L. Ginsberg, and A. J. Parkes. Generalizing Boolean satisfiability I: Background and survey of existing work. JAIR, 21:193-243, 2004.","title":"Generalizing Boolean satisfiability I: Background and survey of existing work","authors":"H E Dixon, M L Ginsberg, A J Parkes","year":"2004","venue":"JAIR","doi":null,"url":null,"internal_url":null},{"position":10,"raw":"[Eén and Sörensson, 2006] N. Eén and N. Sörensson. Translating pseudo-Boolean constraints into SAT. JSAT, 2(1-4):1-26, 2006.","title":"Translating pseudo-Boolean constraints into SAT","authors":"N Eén, N Sörensson","year":"2006","venue":"JSAT","doi":null,"url":null,"internal_url":null},{"position":11,"raw":"[Haken, 1985] A. Haken. The intractability of resolution. Theor. Comput. Sci., 39(2-3):297-308, 1985.","title":"The intractability of resolution","authors":"A Haken","year":"1985","venue":"Theor. Comput. Sci","doi":null,"url":null,"internal_url":null},{"position":12,"raw":"[Hooker, 1992] J. N. Hooker. Generalized resolution for 0-1 linear inequalities. Annals Math. Artif. Intell., 6(1):271-286, 1992.","title":"Generalized resolution for 0-1 linear inequalities","authors":"J N Hooker","year":"1992","venue":"Annals Math. Artif. Intell","doi":null,"url":null,"internal_url":null},{"position":13,"raw":"[Impagliazzo and Paturi, 2001] R. Impagliazzo and R. Paturi. On the complexity of k-SAT. JCSS, 62(2):367-375, 2001.","title":"On the complexity of k-SAT","authors":"R Impagliazzo, R Paturi","year":"2001","venue":"JCSS","doi":null,"url":null,"internal_url":null},{"position":14,"raw":"[Joshi et al., 2015] S. Joshi, R. Martins, and V. M. Manquinho. Generalized totalizer encoding for pseudo-Boolean constraints. In Proc. 21st Conf. Principles and Practice Constr. Prog. (CP '15), pp. 200-209, 2015.","title":"Generalized totalizer encoding for pseudo-Boolean constraints","authors":"S Joshi, R Martins, V M Manquinho","year":"2015","venue":"Proc. 21st Conf. Principles and Practice Constr. Prog. (CP '15","doi":null,"url":null,"internal_url":null},{"position":15,"raw":"[Jovanovic and de Moura, 2013] D. Jovanovic and L. de Moura. Cutting to the chase solving linear integer arithmetic. J. Auto- mated Reasoning, 51(1):79-108, 2013.","title":"Cutting to the chase solving linear integer arithmetic","authors":"D Jovanovic, L De Moura","year":"2013","venue":"J. Automated Reasoning","doi":null,"url":null,"internal_url":null},{"position":16,"raw":"[Le Berre and Parrain, 2010] D. Le Berre and A. Parrain. The Sat4j library, release 2.2. JSAT, 7:59-64, 2010.","title":"The Sat4j library, release 2.2","authors":"D, Le Berre, A Parrain","year":"2010","venue":"JSAT","doi":null,"url":null,"internal_url":null},{"position":17,"raw":"[Levin, 1973] L. A. Levin. Universal sequential search problems. Problemy peredachi informatsii, 9(3):115-116, 1973.","title":"Universal sequential search problems","authors":"L A Levin","year":"1973","venue":"Problemy peredachi informatsii","doi":null,"url":null,"internal_url":null},{"position":18,"raw":"[Manquinho and Marques-Silva, 2006] V. M. Manquinho and J. Marques-Silva. On using cutting planes in pseudo-Boolean optimization. JSAT, 2:209-219, 2006.","title":"On using cutting planes in pseudo-Boolean optimization","authors":"V M Manquinho, J Marques-Silva","year":"2006","venue":"JSAT","doi":null,"url":null,"internal_url":null},{"position":19,"raw":"[Marques-Silva and Sakallah, 1999] J. P. Marques-Silva and K. A. Sakallah. GRASP: A search algorithm for propositional satisfia- bility. IEEE Trans. Comput., 48(5):506-521, 1999.","title":"GRASP: A search algorithm for propositional satisfiability","authors":"J P Marques-Silva, K A Sakallah","year":"1999","venue":"IEEE Trans. Comput","doi":null,"url":null,"internal_url":null},{"position":20,"raw":"[Martins et al., 2014] R. Martins, V. M. Manquinho, and I. Lynce. Open-WBO: A modular MaxSAT solver. In Proc. 17th Conf. Theory Appl. Sat. Testing (SAT '14), pp. 438-445, 2014.","title":"Open-WBO: A modular MaxSAT solver","authors":"R Martins, V M Manquinho, I Lynce","year":"2014","venue":"Proc. 17th Conf. Theory Appl. Sat. Testing (SAT '14)","doi":null,"url":null,"internal_url":null},{"position":21,"raw":"[Moskewicz et al., 2001] M. W. Moskewicz, C. F. Madigan, Y. Zhao, L. Zhang, and S. Malik. Chaff: Engineering an efficient SAT solver. In Proc. 38th Design Automation Conf. (DAC '01), pp. 530-535, 2001.","title":"Chaff: Engineering an efficient SAT solver","authors":"M W Moskewicz, C F Madigan, Y Zhao, L Zhang, S Malik","year":"2001","venue":"Proc. 38th Design Automation Conf. (DAC '01)","doi":null,"url":null,"internal_url":null},{"position":22,"raw":"[Sakai and Nabeshima, 2015] M. Sakai and H. Nabeshima. Con- struction of an ROBDD for a PB-constraint in band form and related techniques for PB-solvers. IEICE Trans. Info. Syst., 98-D(6):1121-1127, 2015.","title":"Construction of an ROBDD for a PB-constraint in band form and related techniques for PB-solvers","authors":"M Sakai, H Nabeshima","year":"2015","venue":"IEICE Trans. Info. Syst","doi":null,"url":null,"internal_url":null},{"position":23,"raw":"[Sheini and Sakallah, 2006] H. M. Sheini and K. A. Sakallah. Pueblo: A hybrid pseudo-Boolean SAT solver. JSAT, 2(1-4):165-189, 2006.","title":"Pueblo: A hybrid pseudo-Boolean SAT solver","authors":"H M Sheini, K A Sakallah","year":"2006","venue":"JSAT","doi":null,"url":null,"internal_url":null},{"position":24,"raw":"[Urquhart, 1987] A. Urquhart. Hard examples for resolution. J. ACM, 34(1):209-219, 1987.","title":"Hard examples for resolution","authors":"A Urquhart","year":"1987","venue":"J. ACM","doi":null,"url":null,"internal_url":null},{"position":25,"raw":"[Vinyals et al., 2018] M. Vinyals, J. Elffers, J. Giráldez-Cru, S. Gocht, and J. Nordström. In between resolution and cutting planes: A study of proof systems for pseudo-Boolean SAT solv- ing. In Proc. 21st Conf. Theory Appl. Sat. Testing (SAT '18), 2018. To appear.","title":"In between resolution and cutting planes: A study of proof systems for pseudo-Boolean SAT solving","authors":"M Vinyals, J Elffers, J Giráldez-Cru, S Gocht, J Nordström","year":"2018","venue":"Proc. 21st Conf. Theory Appl. Sat. Testing (SAT '18)","doi":null,"url":null,"internal_url":null}],"license_note":"Copyright International Joint Conferences on Artificial Intelligence."}