Best-Case and Worst-Case Sparsifiability of Boolean CSPs

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

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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

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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


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