Document Open Access Logo

Best-Case and Worst-Case Sparsifiability of Boolean CSPs

Authors Hubie Chen, Bart M. P. Jansen, Astrid Pieterse

Thumbnail PDF


  • Filesize: 0.49 MB
  • 13 pages

Document Identifiers

Author Details

Hubie Chen
  • Birkbeck, University of London, Malet Street, Bloomsbury, London WC1E 7HX, United Kingdom
Bart M. P. Jansen
  • Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
Astrid Pieterse
  • Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands

Cite AsGet BibTex

Hubie Chen, Bart M. P. Jansen, and Astrid Pieterse. Best-Case and Worst-Case Sparsifiability of Boolean CSPs. In 13th International Symposium on Parameterized and Exact Computation (IPEC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 115, pp. 15:1-15:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)


We continue the investigation of polynomial-time sparsification for NP-complete Boolean Constraint Satisfaction Problems (CSPs). The goal in sparsification is to reduce the number of constraints in a problem instance without changing the answer, such that a bound on the number of resulting constraints can be given in terms of the number of variables n. We investigate how the worst-case sparsification size depends on the types of constraints allowed in the problem formulation (the constraint language). Two algorithmic results are presented. The first result essentially shows that for any arity k, the only constraint type for which no nontrivial sparsification is possible has exactly one falsifying assignment, and corresponds to logical OR (up to negations). Our second result concerns linear sparsification, that is, a reduction to an equivalent instance with O(n) constraints. Using linear algebra over rings of integers modulo prime powers, we give an elegant necessary and sufficient condition for a constraint type to be captured by a degree-1 polynomial over such a ring, which yields linear sparsifications. The combination of these algorithmic results allows us to prove two characterizations that capture the optimal sparsification sizes for a range of Boolean CSPs. For NP-complete Boolean CSPs whose constraints are symmetric (the satisfaction depends only on the number of 1 values in the assignment, not on their positions), we give a complete characterization of which constraint languages allow for a linear sparsification. For Boolean CSPs in which every constraint has arity at most three, we characterize the optimal size of sparsifications in terms of the largest OR that can be expressed by the constraint language.

Subject Classification

ACM Subject Classification
  • Computing methodologies → Symbolic and algebraic algorithms
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Parameterized complexity and exact algorithms
  • constraint satisfaction problems
  • kernelization
  • sparsification
  • lower bounds


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Hans L. Bodlaender, Bart M. P. Jansen, and Stefan Kratsch. Kernelization Lower Bounds by Cross-Composition. SIAM J. Discrete Math., 28(1):277-305, 2014. URL:
  2. Hans L. Bodlaender, Stéphan Thomassé, and Anders Yeo. Kernel bounds for disjoint cycles and disjoint paths. Theor. Comput. Sci., 412(35):4570-4578, 2011. URL:
  3. Hubie Chen. A Rendezvous of Logic, Complexity, and Algebra. ACM Computing Surveys, 42(1), 2009. URL:
  4. Hubie Chen, Bart M. P. Jansen, and Astrid Pieterse. Best-case and Worst-case Sparsifiability of Boolean CSPs. CoRR, abs/1809.06171v1, 2018. URL:
  5. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL:
  6. Holger Dell and Dániel Marx. Kernelization of packing problems. In Proc. 23rd SODA, pages 68-81, 2012. URL:
  7. Holger Dell and Dieter van Melkebeek. Satisfiability Allows No Nontrivial Sparsification unless the Polynomial-Time Hierarchy Collapses. J. ACM, 61(4):23:1-23:27, 2014. URL:
  8. Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, 2013. URL:
  9. Bart M. P. Jansen. On Sparsification for Computing Treewidth. Algorithmica, 71(3):605-635, 2015. URL:
  10. Bart M. P. Jansen and Astrid Pieterse. Optimal Sparsification for Some Binary CSPs Using Low-Degree Polynomials. In Proc. 41st MFCS, pages 71:1-71:14, 2016. URL:
  11. Bart M. P. Jansen and Astrid Pieterse. Optimal Data Reduction for Graph Coloring Using Low-Degree Polynomials. In Proc. 12th IPEC, pages 22:1-22:12, 2017. URL:
  12. Bart M. P. Jansen and Astrid Pieterse. Sparsification Upper and Lower Bounds for Graph Problems and Not-All-Equal SAT. Algorithmica, 79(1):3-28, 2017. URL:
  13. Bart M. P. Jansen and Astrid Pieterse. Optimal Sparsification for Some Binary CSPs Using Low-Degree Polynomials. CoRR, abs/1606.03233v2, 2018. URL:
  14. Stefan Kratsch, Geevarghese Philip, and Saurabh Ray. Point Line Cover: The Easy Kernel is Essentially Tight. ACM Trans. Algorithms, 12(3):40:1-40:16, 2016. URL:
  15. Victor Lagerkvist and Magnus Wahlström. Kernelization of Constraint Satisfaction Problems: A Study Through Universal Algebra. In Proc. 23rd CP, pages 157-171, 2017. URL:
  16. Daniel Lokshtanov, Neeldhara Misra, and Saket Saurabh. Kernelization - Preprocessing with a Guarantee. In The Multivariate Algorithmic Revolution and Beyond, pages 129-161, 2012. URL:
  17. Lásló Lovász. Chromatic number of hypergraphs and linear algebra. In Studia Scientiarum Mathematicarum Hungarica 11, pages 113-114, 1976. Google Scholar
  18. Thomas J. Schaefer. The Complexity of Satisfiability Problems. In Proc. 10th STOC, pages 216-226, 1978. URL:
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail