{"type":"ScholarlyArticle","title":"Linear Satisfiability Preserving Assignments (Extended Abstract)","authors":[{"name":"Kei Kimura","slug":"kei-kimura","given":"Kei","family":"Kimura","affiliation":"Toyohashi University of Technology"},{"name":"Kazuhisa Makino","slug":"kazuhisa-makino","given":"Kazuhisa","family":"Makino","affiliation":"Kyoto University"}],"abstract":"In this paper, we study several classes of satisfiability preserving assignments to the constraint satisfaction problem. In particular, we consider fixable, autark and satisfying assignments.\r\nSince it is in general NP-hard to find a nontrivial (i.e., nonempty) satisfiability preserving assignment, we introduce linear satisfiability preserving assignments, which are defined by polyhedral cones in an associated vector space. The vector space is obtained by the identification, introduced by Kullmann, of assignments with real vectors. We consider arbitrary polyhedral cones, where only restricted classes of cones for autark assignments are considered in the literature. We reveal that cones in certain classes are maximal as a convex subset of the set of the associated vectors, which can be regarded as extensions of Kullmann's results for autark assignments of CNFs. As algorithmic results, we present a pseudo-polynomial time algorithm that computes a linear fixable assignment for a given integer linear system, which implies the well known pseudo-polynomial solvability for integer linear systems such as two-variable-per-inequality, Horn and q-Horn systems.","doi":"10.24963/ijcai.2018/797","url":"https://ijcai.org/proceedings/2018/797","pdf":"https://ijcai.org/proceedings/2018/0797.pdf","markdown":"https://ijcai.org/proceedings/2018/797.md","bibtex":"https://ijcai.org/proceedings/2018/bibtex/797","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":"Journal track","track":"Journal track","number":797,"pages":{"first":5622,"last":5626},"keywords":["Constraints and SAT: Constraint Satisfaction"],"videos":[],"references":[{"position":1,"raw":"[Bar-Yehuda and Rawitz, 2001] Reuven Bar-Yehuda and Dror Rawitz. Efficient algorithms for integer pro- grams with two variables per constraint. Algorithmica, 29(4):595-609, 2001.","title":"Efficient algorithms for integer programs with two variables per constraint","authors":"Rawitz Bar-Yehuda, Reuven Bar-Yehuda, Dror Rawitz","year":"2001","venue":"Algorithmica","doi":null,"url":null,"internal_url":null},{"position":2,"raw":"[Bordeaux et al., 2008] Lucas Bordeaux, Marco Cadoli, and Toni Mancini. A unifying framework for structural proper- ties of CSPs: Definitions, complexity, tractability. Journal of Artificial Intelligence Research, 32:607-629, 2008.","title":"A unifying framework for structural properties of CSPs: Definitions, complexity, tractability","authors":"Bordeaux","year":"2008","venue":"Journal of Artificial Intelligence Research","doi":null,"url":null,"internal_url":null},{"position":3,"raw":"[Boros et al., 1990] Endre Boros, Yves Crama, and Peter L. Hammer. Polynomial-time inference of all valid implica- tions for Horn and related formulae. Annals of Mathemat- ics and Artificial Intelligence, 1:21-32, 1990.","title":"Polynomial-time inference of all valid implications for Horn and related formulae","authors":"Endre Boros, Yves Crama, Peter L Hammer","year":"1990","venue":"Annals of Mathematics and Artificial Intelligence","doi":null,"url":null,"internal_url":null},{"position":4,"raw":"[Brualdi and Shader, 1995] Richard A. Brualdi and Bryan L. Shader. Matrices of Sign-Solvable Linear Systems. Cam- bridge University Press, Cambridge, 1995.","title":"Matrices of Sign-Solvable Linear Systems","authors":"Richard A Brualdi, Bryan L Shader","year":"1995","venue":null,"doi":null,"url":null,"internal_url":null},{"position":5,"raw":"[Creignou et al., 2001] Nadia Creignou, Sanjeev Khanna, and Madhu Sudan. Complexity Classifications of Boolean Constraint Satisfaction Problems. Society for Industrial and Applied Mathematics, Pennsylvania, USA, 2001.","title":"Complexity Classifications of Boolean Constraint Satisfaction Problems","authors":"Nadia Creignou, Sanjeev Khanna, Madhu Sudan","year":"2001","venue":null,"doi":null,"url":null,"internal_url":null},{"position":6,"raw":"[Davis and Putnam, 1960] Martin Davis and Hilary Putnam. A computing procedure for quantification theory. Journal of the ACM, 7:201-215, 1960.","title":"A computing procedure for quantification theory","authors":"Putnam ; Davis, Martin Davis, Hilary Putnam","year":"1960","venue":"Journal of the ACM","doi":null,"url":null,"internal_url":null},{"position":7,"raw":"[Davis et al., 1962] Martin Davis, George Logemann, and Donald Loveland. A machine program for theorem- proving. Communications of the ACM, 5:394-397, 1962.","title":"A machine program for theoremproving","authors":"Martin Davis, George Logemann, Donald Loveland","year":"1962","venue":"Communications of the ACM","doi":null,"url":null,"internal_url":null},{"position":8,"raw":"[Dechter, 2003] Rina Dechter. Constraint Processing. Mor- gan Kaufmann, California, USA, 2003.","title":"Constraint Processing","authors":"Rina Dechter","year":"2003","venue":null,"doi":null,"url":null,"internal_url":null},{"position":9,"raw":"[Even et al., 1976] Shimon Even, Alon Itai, and Adi Shamir. On the complexity of timetable and multicommodity flow problems. SIAM Journal on Computing, 5(4):691-703, 1976.","title":"On the complexity of timetable and multicommodity flow problems","authors":"Shimon Even, Alon Itai, Adi Shamir","year":"1976","venue":"SIAM Journal on Computing","doi":null,"url":null,"internal_url":null},{"position":10,"raw":"[Garey and Johnson, 1979] Michael R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman and Com- pany, USA, 1979.","title":"Computers and Intractability: A Guide to the Theory of NP-Completeness","authors":"Michael R Garey, David S Johnson","year":"1979","venue":null,"doi":null,"url":null,"internal_url":null},{"position":11,"raw":"[Glover, 1964] Fred Glover. A bound escalation method for the solution of integer linear programs. Cahiers du Cen- tre d'Etudes de Recherche Operationelle, 6(3):131-168, 1964.","title":"A bound escalation method for the solution of integer linear programs","authors":"Fred Glover","year":"1964","venue":"Cahiers du Centre d'Etudes de Recherche Operationelle","doi":null,"url":null,"internal_url":null},{"position":12,"raw":"[Grötschel et al., 1993] Martin Grötschel, László Lovász, and Alexander Schrijver. Geometric Algorithms and Combinatorial Optimization Second Corrected Edition. Springer-Verlag, Heidelberg, Germany, 1993.","title":"Geometric Algorithms and Combinatorial Optimization Second Corrected Edition","authors":"Martin Grötschel, László Lovász, Alexander Schrijver","year":"1993","venue":null,"doi":null,"url":null,"internal_url":null},{"position":13,"raw":"[Henschen and Wos, 1974] Lawrence J. Henschen and Lawrence A. Wos. Unit refutations and Horn sets. Journal of the ACM, 21:590-605, 1974.","title":"Unit refutations and Horn sets","authors":"Lawrence J Henschen, Lawrence A Wos","year":"1974","venue":"Journal of the ACM","doi":null,"url":null,"internal_url":null},{"position":14,"raw":"[Hochbaum et al., 1993] Dorit S. Hochbaum, Nimrod Megiddo, Joseph Naor, and Arie Tamir. Tight bounds and 2-approximation algorithms for integer programs with two variables per inequality. Mathematical Programming, 62:69-83, 1993.","title":"Tight bounds and 2-approximation algorithms for integer programs with two variables per inequality","authors":"Dorit S Hochbaum, Nimrod Megiddo, Joseph Naor, Arie Tamir","year":"1993","venue":"Mathematical Programming","doi":null,"url":null,"internal_url":null},{"position":15,"raw":"[Jonsson and Krokhin, 2004] Peter Jonsson and Andrei Krokhin. Recognizing frozen variables in constraint satisfaction problems. Theoretical Computer Science, 329:93-113, 2004.","title":"Recognizing frozen variables in constraint satisfaction problems","authors":"Jonsson, Krokhin, Peter Jonsson, Andrei Krokhin","year":"2004","venue":"Theoretical Computer Science","doi":null,"url":null,"internal_url":null},{"position":16,"raw":"[Kimura and Makino, 2016] Kei Kimura and Kazuhisa Makino. Trichotomy for integer linear systems based on their sign patterns. Discrete Applied Mathematics, 200:67-78, 2016.","title":"Trichotomy for integer linear systems based on their sign patterns","authors":"Makino ; ] Kei Kimura, Kazuhisa Kimura, Makino","year":"2016","venue":"Discrete Applied Mathematics","doi":null,"url":null,"internal_url":null},{"position":17,"raw":"[Kimura and Makino, 2018] Kei Kimura and Kazuhisa Makino. Linear satisfiability preserving assignments. Journal of Artificial Intelligence Research, 61:291-321, 2018.","title":"Linear satisfiability preserving assignments","authors":"Makino ; ] Kei Kimura, Kazuhisa Kimura, Makino","year":"2018","venue":"Journal of Artificial Intelligence Research","doi":null,"url":null,"internal_url":null},{"position":18,"raw":"[Kleine Büning and Kullmann, 2009] Hans Kleine Büning and Oliver Kullmann. Minimal unsatisfiability and au- tarkies. In Armin Biere, Marijn Heule, Hans van Maaren, and Toby Walsh, editors, Handbook of Satisfiability, pages 339-401. IOS Press, Amsterdam, 2009.","title":"Minimal unsatisfiability and autarkies","authors":"Kleine Büning, ; Kullmann, Kleine Hans, Oliver Büning, Kullmann","year":"2009","venue":"Handbook of Satisfiability","doi":null,"url":null,"internal_url":null},{"position":19,"raw":"[Kullmann, 2000] Oliver Kullmann. Investigations on autark assignments. Discrete Applied Mathematics, 107:99-137, 2000.","title":"Investigations on autark assignments","authors":"Oliver Kullmann, Kullmann","year":"2000","venue":"Discrete Applied Mathematics","doi":null,"url":null,"internal_url":null},{"position":20,"raw":"[Kullmann, 2003] Oliver Kullmann. Lean clause-sets: Gen- eralizations of minimally unsatisfiable clause-sets. Dis- crete Applied Mathematics, 130:209-249, 2003.","title":"Lean clause-sets: Generalizations of minimally unsatisfiable clause-sets","authors":"Oliver Kullmann","year":"2003","venue":"Discrete Applied Mathematics","doi":null,"url":null,"internal_url":null},{"position":21,"raw":"[Lewis, 1978] Harry R. Lewis. Renaming a set of clauses as a Horn set. Journal of the ACM, 25:134-135, 1978.","title":"Renaming a set of clauses as a Horn set","authors":"R Harry, Lewis","year":"1978","venue":"Journal of the ACM","doi":null,"url":null,"internal_url":null},{"position":22,"raw":"[Monasson et al., 1999] Rémi Monasson, Riccardo Zecchina, Scott Kirkpatrick, Bart Selman, and Lidror Troyansky. Determining computational complexity from characteristic 'phase transitions'. Nature, 400:133-137, 1999.","title":"Determining computational complexity from characteristic 'phase transitions","authors":"Rémi Monasson, Riccardo Zecchina, Scott Kirkpatrick, Bart Selman, Lidror Troyansky","year":"1999","venue":"Nature","doi":null,"url":null,"internal_url":null},{"position":23,"raw":"[Monien and Speckenmeyer, 1985] B. Monien and E. Speckenmeyer. Solving satisfaiability in less than 2 n steps. Discrete Applied Mathematics, 10:287-295, 1985.","title":"Solving satisfaiability in less than 2 n steps","authors":"B Monien, E Speckenmeyer","year":"1985","venue":"Discrete Applied Mathematics","doi":null,"url":null,"internal_url":null},{"position":24,"raw":"[Petke and Jeavons, 2011] Justyna Petke and Peter Jeavons. The order encoding: From tractable CSP to tractable SAT. In Proceedings of the 14th International Conference on Theory and Applications of Satisfiability Testing, pages 371-372, 2011.","title":"The order encoding: From tractable CSP to tractable SAT","authors":"Jeavons Petke, Justyna Petke, Peter Jeavons","year":"2011","venue":"Proceedings of the 14th International Conference on Theory and Applications of Satisfiability Testing","doi":null,"url":null,"internal_url":null},{"position":25,"raw":"[Rossi et al., 2006] Francesca Rossi, Peter van Beek, and Toby Walsh, editors. Handbook of Constraint Program- ming. Elsevier Science, New York, USA, 2006.","title":"Handbook of Constraint Programming","authors":null,"year":"2006","venue":null,"doi":null,"url":null,"internal_url":null},{"position":26,"raw":"[Tamura et al., 2009] Naoyuki Tamura, Akiko Taga, Satoshi Kitagawa, and Mutsunori Banbara. Compiling finite linear CSP into SAT. Constraints, 14:254-272, 2009.","title":"Compiling finite linear CSP into SAT","authors":"Naoyuki Tamura, Akiko Taga, Satoshi Kitagawa, Mutsunori Banbara","year":"2009","venue":"Constraints","doi":null,"url":null,"internal_url":null},{"position":27,"raw":"[Tsang, 1993] Edward Tsang. Foundations of Constraint Satisfaction. Academic Pr, London and San Diego, 1993.","title":"Foundations of Constraint Satisfaction","authors":"Edward Tsang","year":"1993","venue":null,"doi":null,"url":null,"internal_url":null},{"position":28,"raw":"[van Maaren and Dang, 2002] Hans van Maaren and Chuangyin Dang. Simplicial pivoting algorithms for a tractable class of integer programs. Journal of Combina- torial Optimization, 6(2):133-142, 2002.","title":"Simplicial pivoting algorithms for a tractable class of integer programs","authors":"Van Maaren, Chuangyin Dang ; Hans Van Maaren, Dang","year":"2002","venue":"Journal of Combinatorial Optimization","doi":null,"url":null,"internal_url":null},{"position":29,"raw":"[van Maaren, 2000] Hans van Maaren. A short note on some tractable cases of the satisfiability problem. Information and Computation, 158:125-130, 2000.","title":"A short note on some tractable cases of the satisfiability problem","authors":"; Van Maaren, Hans Van Maaren","year":"2000","venue":"Information and Computation","doi":null,"url":null,"internal_url":null}],"license_note":"Copyright International Joint Conferences on Artificial Intelligence."}