{"type":"ScholarlyArticle","title":"Fast Algorithms for SAT with Bounded Occurrences of Variables","authors":[{"name":"Junqiang Peng","slug":"junqiang-peng","given":"Junqiang","family":"Peng","affiliation":"University of Electronic Science and Technology of China , Chengdu , China"},{"name":"Mingyu Xiao","slug":"mingyu-xiao","given":"Mingyu","family":"Xiao","affiliation":"University of Electronic Science and Technology of China , Chengdu , China"}],"abstract":"We present fast algorithms for the general CNF satisfiability problem (SAT) with running-time bound O*({c_d}^n), where c_d is a function of the maximum occurrence d of variables (d can also be the average occurrence when each variable appears at least twice), and n is the number of variables in the input formula. Similar to SAT with bounded clause lengths, SAT with bounded occurrences of variables has also been extensively studied in the literature. Especially, the running-time bounds for small values of d, such as d=3 and d=4, have become bottlenecks for algorithms evaluated by the formula length L and other algorithms. In this paper, we show that SAT can be solved in time O*(1.1238^n) for d=3 and O*(1.2628^n) for d=4, improving the previous results O*(1.1279^n) and O*(1.2721^n) obtained by Wahlström (SAT 2005) nearly 20 years ago. For d>=5, we obtain a running time bound of O*(1.0641^{dn}), implying a bound of O*(1.0641^L) with respect to the formula length L, which is also a slight improvement over the previous bound.","doi":"10.24963/ijcai.2023/223","url":"https://ijcai.org/proceedings/2023/223","pdf":"https://ijcai.org/proceedings/2023/0223.pdf","markdown":"https://ijcai.org/proceedings/2023/223.md","bibtex":"https://ijcai.org/proceedings/2023/bibtex/223","volume":{"slug":"2023","title":"Proceedings of the Thirty-Second International Joint Conference on Artificial Intelligence","short_title":"IJCAI 2023","url":"https://ijcai.org/proceedings/2023/","year":2023,"location":"Macao, SAR","dates":"19-25 August 2023","editors":"Edith Elkind","isbn":"978-1-956792-03-4","publisher":"International Joint Conferences on Artificial Intelligence Organization","doi_prefix":"10.24963/ijcai.2023"},"section":"Constraint Satisfaction and Optimization","track":"Main Track","number":223,"pages":{"first":2004,"last":2012},"keywords":["Constraint Satisfaction and Optimization: CSO: Satisfiabilty"],"videos":[],"references":[{"position":1,"raw":"[Alferov and Bliznets, 2021] Vasily Alferov and Ivan Bliznets. New length dependent algorithm for maximum satisfiability problem. In Thirty-Fifth AAAI Conference on Artificial Intelligence, AAAI 2021, Thirty-Third Confer- ence on Innovative Applications of Artificial Intelligence, IAAI 2021, The Eleventh Symposium on Educational Advances in Artificial Intelligence, EAAI 2021, Virtual Event, February 2-9, 2021, pages 3634-3641. AAAI Press, 2021.","title":"New length dependent algorithm for maximum satisfiability problem","authors":"Bliznets Alferov, Vasily Alferov, Ivan Bliznets","year":"2021","venue":"Thirty-Fifth AAAI Conference on Artificial Intelligence, AAAI 2021, Thirty-Third Conference on Innovative Applications of Artificial Intelligence, IAAI 2021, The Eleventh Symposium on Educational Advances in Artificial Intelligence, EAAI 2021, Virtual Event","doi":null,"url":null,"internal_url":null},{"position":2,"raw":"[Bansal and Raman, 1999] Nikhil Bansal and Venkatesh Ra- man. Upper bounds for maxsat: Further improved. In Alok Aggarwal and C. Pandu Rangan, editors, Algorithms and Computation, 10th International Symposium, ISAAC '99, Chennai, India, December 16-18, 1999, Proceedings, volume 1741 of Lecture Notes in Computer Science, pages 247-258. Springer, 1999.","title":"Upper bounds for maxsat: Further improved","authors":"Nikhil Bansal, Venkatesh Raman","year":"1999","venue":"Algorithms and Computation, 10th International Symposium, ISAAC '99","doi":null,"url":null,"internal_url":null},{"position":3,"raw":"[Beigel and Eppstein, 2005] Richard Beigel and David Epp- stein. 3-coloring in time O(1.3289 n ). J. Algorithms, 54(2):168-204, 2005.","title":"3-coloring in time O(1.3289 n )","authors":"Richard Beigel, David Eppstein","year":"2005","venue":"J. Algorithms","doi":null,"url":null,"internal_url":null},{"position":4,"raw":"[Belova and Bliznets, 2020] Tatiana Belova and Ivan Bliznets. Algorithms for (n, 3)-maxsat and parameteriza- tion above the all-true assignment. Theor. Comput. Sci., 803:222-233, 2020.","title":"Algorithms for (n, 3)-maxsat and parameterization above the all-true assignment","authors":"Tatiana Belova, Ivan Bliznets","year":"2020","venue":"Theor. Comput. Sci","doi":null,"url":null,"internal_url":null},{"position":5,"raw":"[Bliznets, 2013] Ivan A Bliznets. A new upper bound for (n, 3)-max-sat. Journal of Mathematical Sciences, 188(1):1- 6, 2013.","title":"A new upper bound for (n, 3)-max-sat","authors":"A Ivan, Bliznets","year":"2013","venue":"Journal of Mathematical Sciences","doi":null,"url":null,"internal_url":null},{"position":6,"raw":"[Brilliantov et al., 2023] Kirill Brilliantov, Vasily Alferov, and Ivan Bliznets. Improved algorithms for maximum sat- isfiability and its special cases. In Thirty-Seventh AAAI Conference on Artificial Intelligence, AAAI 2023, 2023. (to appear).","title":"Improved algorithms for maximum satisfiability and its special cases","authors":"Kirill Brilliantov, Vasily Alferov, Ivan Bliznets","year":"2023","venue":"Thirty-Seventh AAAI Conference on Artificial Intelligence, AAAI 2023","doi":null,"url":null,"internal_url":null},{"position":7,"raw":"[Chen and Liu, 2009] Jianer Chen and Yang Liu. An im- proved SAT algorithm in terms of formula length. In Frank K. H. A. Dehne, Marina L. Gavrilova, Jörg-Rüdiger Sack, and Csaba D. Tóth, editors, Algorithms and Data Struc- tures, 11th International Symposium, WADS 2009, Banff, Canada, August 21-23, 2009. Proceedings, volume 5664 of Lecture Notes in Computer Science, pages 144-155. Springer, 2009.","title":"An improved SAT algorithm in terms of formula length","authors":"Chen, Liu ; ] Jianer Chen, Yang Liu","year":"2009","venue":"Algorithms and Data Structures, 11th International Symposium, WADS 2009","doi":null,"url":null,"internal_url":null},{"position":8,"raw":"[Chu et al., 2021] Huairui Chu, Mingyu Xiao, and Zhe Zhang. An improved upper bound for SAT. In Thirty-Fifth AAAI Conference on Artificial Intelligence, AAAI 2021, Thirty-Third Conference on Innovative Applications of Ar- tificial Intelligence, IAAI 2021, The Eleventh Symposium on Educational Advances in Artificial Intelligence, EAAI 2021, Virtual Event, February 2-9, 2021, pages 3707- 3714. AAAI Press, 2021.","title":"An improved upper bound for SAT","authors":"Chu","year":"2021","venue":"Thirty-Fifth AAAI Conference on Artificial Intelligence, AAAI 2021, Thirty-Third Conference on Innovative Applications of Artificial Intelligence, IAAI 2021, The Eleventh Symposium on Educational Advances in Artificial Intelligence, EAAI 2021, Virtual Event","doi":null,"url":null,"internal_url":null},{"position":9,"raw":"[Cook, 1971] Stephen A. Cook. The complexity of theorem- proving procedures. In Michael A. Harrison, Ranan B. Banerji, and Jeffrey D. Ullman, editors, Proceedings of the 3rd Annual ACM Symposium on Theory of Comput- ing, May 3-5, 1971, Shaker Heights, Ohio, USA, pages 151-158. ACM, 1971.","title":"The complexity of theoremproving procedures","authors":"Stephen A Cook","year":"1971","venue":"Proceedings of the 3rd Annual ACM Symposium on Theory of Computing","doi":null,"url":null,"internal_url":null},{"position":10,"raw":"[Dantsin et al., 2004] Evgeny Dantsin, Edward A. Hirsch, and Alexander Wolpert. Algorithms for SAT based on search in hamming balls. In Volker Diekert and Michel Habib, editors, STACS 2004, 21st Annual Symposium on Theoretical Aspects of Computer Science, Montpellier, France, March 25-27, 2004, Proceedings, volume 2996 of Lecture Notes in Computer Science, pages 141-151. Springer, 2004.","title":"Algorithms for SAT based on search in hamming balls","authors":"Evgeny Dantsin, Edward A Hirsch, Alexander Wolpert","year":"2004","venue":"STACS 2004, 21st Annual Symposium on Theoretical Aspects of Computer Science","doi":null,"url":null,"internal_url":null},{"position":11,"raw":"[Dantsin et al., 2006] Evgeny Dantsin, Edward A. Hirsch, and Alexander Wolpert. Clause shortening combined with pruning yields a new upper bound for deterministic SAT algorithms. In Tiziana Calamoneri, Irene Finocchi, and Giuseppe F. Italiano, editors, Algorithms and Complexity, 6th Italian Conference, CIAC 2006, Rome, Italy, May 29- 31, 2006, Proceedings, volume 3998 of Lecture Notes in Computer Science, pages 60-68. Springer, 2006.","title":"Clause shortening combined with pruning yields a new upper bound for deterministic SAT algorithms","authors":"Evgeny Dantsin, Edward A Hirsch, Alexander Wolpert","year":"2006","venue":"Algorithms and Complexity, 6th Italian Conference, CIAC 2006","doi":null,"url":null,"internal_url":null},{"position":12,"raw":"[Davis and Putnam, 1960] Martin Davis and Hilary Putnam. A computing procedure for quantification theory. J. ACM, 7(3):201-215, 1960.","title":"A computing procedure for quantification theory","authors":"Martin Davis, Hilary Putnam","year":"1960","venue":"J. ACM","doi":null,"url":null,"internal_url":null},{"position":13,"raw":"[Fomin and Kratsch, 2010] Fedor V. Fomin and Dieter Kratsch. Exact Exponential Algorithms. Texts in The- oretical Computer Science. An EATCS Series. Springer, 2010.","title":"Exact Exponential Algorithms","authors":"V Fedor, Dieter Fomin, Kratsch","year":"2010","venue":"Texts in Theoretical Computer Science. An EATCS Series","doi":null,"url":null,"internal_url":null},{"position":14,"raw":"[Fomin et al., 2009] Fedor V. Fomin, Fabrizio Grandoni, and Dieter Kratsch. A measure & conquer approach for the analysis of exact algorithms. J. ACM, 56(5):25:1-25:32, 2009.","title":"A measure & conquer approach for the analysis of exact algorithms","authors":"V Fedor, Fabrizio Fomin, Dieter Grandoni, Kratsch","year":"2009","venue":"J. ACM","doi":null,"url":null,"internal_url":null},{"position":15,"raw":"[Impagliazzo and Paturi, 2001] Russell Impagliazzo and Ra- mamohan Paturi. On the complexity of k-SAT. J. Comput. Syst. Sci., 62(2):367-375, 2001.","title":"Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-SAT","authors":"Paturi Impagliazzo","year":"2001","venue":"J. Comput. Syst. Sci","doi":null,"url":null,"internal_url":null},{"position":16,"raw":"[Kulikov and Kutskov, 2009] A. S. Kulikov and K. Kutskov. New upper bounds for the problem of maximal satisfiabil- ity. Discret. Appl. Math., 19(2):155-172, 2009.","title":"New upper bounds for the problem of maximal satisfiability","authors":"A S Kulikov, K Kutskov","year":"2009","venue":"Discret. Appl. Math","doi":null,"url":null,"internal_url":null},{"position":17,"raw":"[Liu, 2018] Sixue Liu. Chain, generalization of covering code, and deterministic algorithm for k-SAT. In Ioannis Chatzigiannakis, Christos Kaklamanis, Dániel Marx, and Donald Sannella, editors, 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018, July 9-13, 2018, Prague, Czech Republic, volume 107 of LIPIcs, pages 88:1-88:13. Schloss Dagstuhl -Leibniz- Zentrum für Informatik, 2018.","title":"Chain, generalization of covering code, and deterministic algorithm for k-SAT","authors":"Sixue Liu","year":"2018","venue":"45th International Colloquium on Automata, Languages, and Programming, ICALP 2018","doi":null,"url":null,"internal_url":null},{"position":18,"raw":"[Oliver and Luckhardt, 1997] Kullmann Oliver and Horst Luckhardt. Deciding propositional tautologies: Algo- rithms and their complexity. Preprint, pages 1-82, 1997.","title":"Deciding propositional tautologies: Algorithms and their complexity","authors":"Luckhardt ; Oliver, Kullmann Oliver, Horst Luckhardt","year":"1997","venue":null,"doi":null,"url":null,"internal_url":null},{"position":19,"raw":"[Paturi et al., 1999] Ramamohan Paturi, Pavel Pudlák, and Francis Zane. Satisfiability coding lemma. Chic. J. Theor. Comput. Sci., 1999, 1999.","title":"Satisfiability coding lemma","authors":"Paturi","year":"1999","venue":"Chic. J. Theor. Comput. Sci","doi":null,"url":null,"internal_url":null},{"position":20,"raw":"[Paturi et al., 2005] Ramamohan Paturi, Pavel Pudlák, Michael E. Saks, and Francis Zane. An improved exponential-time algorithm for k-SAT. J. ACM, 52(3):337-364, 2005.","title":"An improved exponential-time algorithm for k-SAT","authors":"Paturi","year":"2005","venue":"J. ACM","doi":null,"url":null,"internal_url":null},{"position":21,"raw":"[Peng and Xiao, 2021] Junqiang Peng and Mingyu Xiao. A fast algorithm for SAT in terms of formula length. In Chu- Min Li and Felip Manyà, editors, Theory and Applications of Satisfiability Testing -SAT 2021 -24th International Conference, Barcelona, Spain, July 5-9, 2021, Proceed- ings, volume 12831 of Lecture Notes in Computer Science, pages 436-452. Springer, 2021.","title":"A fast algorithm for SAT in terms of formula length","authors":"Xiao ; Peng, Junqiang Peng, Mingyu Xiao","year":"2021","venue":"Theory and Applications of Satisfiability Testing -SAT 2021 -24th International Conference","doi":null,"url":null,"internal_url":null},{"position":22,"raw":"[Raman et al., 1998] Venkatesh Raman, Bala Ravikumar, and S. Srinivasa Rao. A simplified np-complete MAXSAT problem. Inf. Process. Lett., 65(1):1-6, 1998.","title":"A simplified np-complete MAXSAT problem","authors":"Bala Venkatesh Raman, S Srinivasa Ravikumar, Rao","year":"1998","venue":"Inf. Process. Lett","doi":null,"url":null,"internal_url":null},{"position":23,"raw":"[Scheder, 2021] Dominik Scheder. PPSZ is better than you think. In 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021, Denver, CO, USA, Febru- ary 7-10, 2022, pages 205-216. IEEE, 2021.","title":"PPSZ is better than you think","authors":"Dominik Scheder","year":"2021","venue":"IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021","doi":null,"url":null,"internal_url":null},{"position":24,"raw":"[Schöning, 1999] Uwe Schöning. A probabilistic algorithm for k-SAT and constraint satisfaction problems. In 40th Annual Symposium on Foundations of Computer Science, FOCS '99, 17-18 October, 1999, New York, NY, USA, pages 410-414. IEEE Computer Society, 1999.","title":"A probabilistic algorithm for k-SAT and constraint satisfaction problems","authors":"Uwe Schöning","year":"1999","venue":"40th Annual Symposium on Foundations of Computer Science, FOCS '99","doi":null,"url":null,"internal_url":null},{"position":25,"raw":"[Schuler, 2005] Rainer Schuler. An algorithm for the satis- fiability problem of formulas in conjunctive normal form. J. Algorithms, 54(1):40-44, 2005.","title":"An algorithm for the satisfiability problem of formulas in conjunctive normal form","authors":"Rainer Schuler","year":"2005","venue":"J. Algorithms","doi":null,"url":null,"internal_url":null},{"position":26,"raw":"[Tovey, 1984] Craig A. Tovey. A simplified np-complete satisfiability problem. Discret. Appl. Math., 8(1):85-89, 1984.","title":"A simplified np-complete satisfiability problem","authors":"Craig A Tovey","year":"1984","venue":"Discret. Appl. Math","doi":null,"url":null,"internal_url":null},{"position":27,"raw":"[Wahlström, 2005a] Magnus Wahlström. An algorithm for the SAT problem for formulae of linear length. In Gerth Stølting Brodal and Stefano Leonardi, editors, Al- gorithms -ESA 2005, 13th Annual European Symposium, Palma de Mallorca, Spain, October 3-6, 2005, Proceed- ings, volume 3669 of Lecture Notes in Computer Science, pages 107-118. Springer, 2005.","title":"An algorithm for the SAT problem for formulae of linear length","authors":"Magnus Wahlström, Wahlström","year":"2005","venue":"Algorithms -ESA 2005, 13th Annual European Symposium","doi":null,"url":null,"internal_url":null},{"position":28,"raw":"[Wahlström, 2005b] Magnus Wahlström. Faster exact solv- ing of SAT formulae with a low number of occurrences per variable. In Fahiem Bacchus and Toby Walsh, editors, Theory and Applications of Satisfiability Testing, 8th In- ternational Conference, SAT 2005, St. Andrews, UK, June 19-23, 2005, Proceedings, volume 3569 of Lecture Notes in Computer Science, pages 309-323. Springer, 2005.","title":"Faster exact solving of SAT formulae with a low number of occurrences per variable","authors":"Magnus Wahlström, Wahlström","year":"2005","venue":"Theory and Applications of Satisfiability Testing, 8th International Conference, SAT 2005","doi":null,"url":null,"internal_url":null},{"position":29,"raw":"[Xiao, 2022] Mingyu Xiao. An exact maxsat algorithm: Fur- ther observations and further improvements. In Luc De Raedt, editor, Proceedings of the Thirty-First International Joint Conference on Artificial Intelligence, IJCAI 2022, Vienna, Austria, 23-29 July 2022, pages 1887-1893. ij- cai.org, 2022.","title":"An exact maxsat algorithm: Further observations and further improvements","authors":"Mingyu Xiao","year":"2022","venue":"Proceedings of the Thirty-First International Joint Conference on Artificial Intelligence, IJCAI 2022","doi":null,"url":null,"internal_url":"/proceedings/2022/262"},{"position":30,"raw":"[Xu et al., 2019] Chao Xu, Jianer Chen, and Jianxin Wang. Resolution and linear CNF formulas: Improved (n, 3)- maxsat algorithms. Theor. Comput. Sci., 774:113-123, 2019.","title":"Resolution and linear CNF formulas: Improved (n, 3)maxsat algorithms","authors":"Chao Xu, Jianer Chen, Jianxin Wang","year":"2019","venue":"Theor. Comput. Sci","doi":null,"url":null,"internal_url":null},{"position":31,"raw":"[Xu et al., 2021] Chao Xu, Wenjun Li, Jianxin Wang, and Yongjie Yang. An improved algorithm for the (n, 3)- maxsat problem: asking branchings to satisfy the clauses. J. Comb. Optim., 42(3):524-542, 2021.","title":"An improved algorithm for the (n, 3)maxsat problem: asking branchings to satisfy the clauses","authors":"Chao Xu, Wenjun Li, Jianxin Wang, Yongjie Yang","year":"2021","venue":"J. Comb. Optim","doi":null,"url":null,"internal_url":null}],"license_note":"Copyright International Joint Conferences on Artificial Intelligence."}