Generating Difficult CNF Instances in Unexplored Constrainedness Regions
Article No.: 1.6, Pages 1 - 12
Abstract
When creating benchmarks for satisfiability (SAT) solvers, we need Conjunctive Normal Form (CNF) instances that are easy to build but hard to solve. A recent development in the search for such methods has led to the Balanced SAT algorithm, which can create k-CNF instances with m clauses of high difficulty, for arbitrary k and m. In this article, we introduce the No-Triangle CNF algorithm, a CNF instance generator based on the cluster coefficient graph statistic. We empirically compare the two algorithms by fixing the arity and the number of variables, but varying the number of clauses. We find that the hardest instances produced by each method belong to different constrainedness regions. In the 3-CNF case, for example, hard No-Triangle CNF instances reside in the highly-constrained region (many clauses), while Balanced SAT instances obtained from the same parameters are easy to solve. This allows us to generate difficult instances where existing algorithms fail to do so.
References
[1]
Dimitris Achlioptas. 2009. Random satisfiability. In Handbook of Satisfiability, Armin Biere, Marijn Heule, Hans van Maaren, and Toby Walsh (Eds.). Frontiers in Artificial Intelligence and Applications, Vol. 185. IOS Press, 245--270.
[2]
Giovanni Amendola, Francesco Ricca, and Miroslaw Truszczynski. 2017. Generating hard random Boolean formulas and disjunctive logic programs. In Proceedings of the 26th International Joint Conference on Artificial Intelligence, IJCAI 2017, Melbourne, Australia, August 19--25, 2017, Carles Sierra (Ed.). ijcai.org, 532--538.
[3]
Gilles Audemard and Laurent Simon. 2018. On the glucose SAT solver. International Journal on Artificial Intelligence Tools 27, 1 (2018), 1--25.
[4]
Adrian Balint. 2018. Random k-SAT q-planted solutions. In Proceedings of SAT Competition 2018—Solver and Benchmark Descriptions (Department of Computer Science Series of Publications B), Marijn Heule, Matti Järvisalo, and Martin Suda (Eds.), Vol. B-2018-1. University of Helsinki, 64.
[5]
Tomás Balyo and Lukás Chrpa. 2018. Using algorithm configuration tools to generate hard SAT benchmarks. In Proceedings of the 11th International Symposium on Combinatorial Search, SOCS 2018, Stockholm, Sweden—14--15 July 2018, Vadim Bulitko and Sabine Storandt (Eds.). AAAI Press, 133--137. https://aaai.org/ocs/index.php/SOCS/SOCS18/paper/view/17952.
[6]
Armin Biere. 2018. CaDiCaL, Lingeling, Plingeling, Treengeling and YalSAT entering the SAT competition 2018. In Proc. of SAT Competition 2018—Solver and Benchmark Descriptions (Department of Computer Science Series of Publications B), Marijn Heule, Matti Järvisalo, and Martin Suda (Eds.), Vol. B-2018-1. University of Helsinki, 13--14.
[7]
Peter C. Cheeseman, Bob Kanefsky, and William M. Taylor. 1991. Where the really hard problems are. In Proceedings of the 12th International Joint Conference on Artificial Intelligence. Sydney, Australia, August 24--30, 1991, John Mylopoulos and Raymond Reiter (Eds.). Morgan Kaufmann, 331--340. http://ijcai.org/Proceedings/91-1/Papers/052.pdf.
[8]
Stephen A. Cook. 1971. The complexity of theorem-proving procedures. In Proceedings of the 3rd Annual ACM Symposium on Theory of Computing, May 3-5, 1971, Shaker Heights, Ohio, Michael A. Harrison, Ranan B. Banerji, and Jeffrey D. Ullman (Eds.). ACM, 151--158.
[9]
Martin C. Cooper and Stanislav Zivny. 2011. Tractable triangles. In Proceedings of the 17th International Conference of Principles and Practice of Constraint Programming,CP 2011, Perugia, Italy, September 12--16, 2011. (Lecture Notes in Computer Science), Jimmy Ho-Man Lee (Ed.), Vol. 6876. Springer, 195--209.
[10]
James M. Crawford and Larry D. Auton. 1996. Experimental results on the crossover point in random 3-SAT. Artificial Intelligence 81, 1--2 (1996), 31--57.
[11]
Matti Järvisalo, Daniel Le Berre, Olivier Roussel, and Laurent Simon. 2012. The international SAT solver competitions. AI Magazine 33, 1 (2012). http://www.aaai.org/ojs/index.php/aimagazine/article/view/2395.
[12]
Haixia Jia, Cristopher Moore, and Doug Strain. 2007. Generating hard satisfiable formulas by hiding solutions deceptively. Journal of Artificial Intelligence Research 28 (2007), 107--118.
[13]
Richard M. Karp. 1972. Reducibility among combinatorial problems. In Proceedings of the Symposium on the Complexity of Computer Computations, March 20--22, 1972, IBM Thomas J. Watson Research Center, Yorktown Heights, New York (The IBM Research Symposia Series), Raymond E. Miller and James W. Thatcher (Eds.). Plenum Press, New York, 85--103. http://www.cs.berkeley.edu/%7Eluca/cs172/karp.pdf.
[14]
Donald E. Knuth. 2015. The Art of Computer Programming, Volume 4, Fascicle 6: Satisfiability (1st ed.). Addison-Wesley Professional.
[15]
Stephan Mertens, Marc Mézard, and Riccardo Zecchina. 2006. Threshold values of random K-SAT from the cavity method. Random Structured Algorithms 28, 3 (2006), 340--373.
[16]
David G. Mitchell, Bart Selman, and Hector J. Levesque. 1992. Hard and easy distributions of SAT problems. In Proceedings of the 10th National Conference on Artificial Intelligence, San Jose, CA, July 12--16, 1992, William R. Swartout (Ed.). AAAI Press/The MIT Press, 459--465. http://www.aaai.org/Library/AAAI/1992/aaai92-071.php.
[17]
Alexander Nadel and Vadim Ryvchin. 2018. Chronological backtracking. In Proceedings of the 21st International Conference on Theory and Applications of Satisfiability Testing, SAT 2018 (Held as Part of the Federated Logic Conference, FloC 2018, Oxford, UK, July 9--12, 2018, Lecture Notes in Computer Science), Olaf Beyersdorff and Christoph M. Wintersteiger (Eds.), Vol. 10929. Springer, 111--121.
[18]
Thomas J. Schaefer. 1978. The complexity of satisfiability problems. In Proceedings of the 10th Annual ACM Symposium on Theory of Computing, May 1--3, 1978, San Diego, California, Richard J. Lipton, Walter A. Burkhard, Walter J. Savitch, Emily P. Friedman, and Alfred V. Aho (Eds.). ACM, 216--226.
[19]
Ivor Spence. 2017. Balanced random SAT benchmarks. In Proceedings of SAT Competition 2017 -- Solver and Benchmark Descriptions (Department of Computer Science Series of Publications B), Tomáš Balyo, Marijn Heule, and Matti Järvisalo (Eds.), Vol. B-2017-1. University of Helsinki, 53--54.
[20]
Ivor T. A. Spence. 2015. Weakening cardinality constraints creates harder satisfiability benchmarks. ACM Journal of Experimental Algorithmics 20 (2015), 1.4:1--1.4:14.
[21]
Duncan J. Watts and Steven H. Strogatz. 1998. Collective dynamics of “small-world” networks. Nature 393 (1998), 440--442.
Index Terms
- Generating Difficult CNF Instances in Unexplored Constrainedness Regions
Recommendations
Generating Instances for MAX2SAT with Optimal Solutions
A test instance generator (an instance generator for short) for MAX2SAT is a procedure that produces, given a number n of variables, a 2-CNF formula F of n variables (randomly chosen from some reasonably large domain), and simultaneously provides one of ...
On the Parameterized Complexity of Finding Small Unsatisfiable Subsets of CNF Formulas and CSP Instances
In many practical settings it is useful to find a small unsatisfiable subset of a given unsatisfiable set of constraints. We study this problem from a parameterized complexity perspective, taking the size of the unsatisfiable subset as the natural ...
Beyond CNF: A Circuit-Based QBF Solver
SAT '09: Proceedings of the 12th International Conference on Theory and Applications of Satisfiability TestingState-of-the-art solvers for Quantified Boolean Formulas (QBF) have employed many techniques from the field of Boolean Satisfiability (SAT) including the use of Conjunctive Normal Form (CNF) in representing the QBF formula. Although CNF has worked well ...
Comments
Information & Contributors
Information
Published In
Copyright © 2020 ACM.
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]
Publisher
Association for Computing Machinery
New York, NY, United States
Publication History
Published: 03 May 2020
Accepted: 01 February 2020
Revised: 01 December 2019
Received: 01 September 2019
Published in JEA Volume 25
Author Tags
Qualifiers
- Research-article
- Research
- Refereed
Funding Sources
Contributors
Other Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
- 0Total Citations
- 678Total Downloads
- Downloads (Last 12 months)275
- Downloads (Last 6 weeks)25
Reflects downloads up to 03 Sep 2024
Other Metrics
Citations
View Options
View options
View or Download as a PDF file.
PDFeReader
View online with eReader.
eReaderHTML Format
View this article in HTML Format.
HTML FormatGet Access
Login options
Check if you have access through your login credentials or your institution to get full access on this article.
Sign in