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Best-Case and Worst-Case Sparsifiability of Boolean CSPs

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Abstract

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—and identify constraint languages giving the best-possible and worst-possible behavior for worst-case sparsifiability. 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.

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Notes

  1. It is possible [18, 24] to achieve \(d=2\), although this will not be necessary for our purposes.

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Acknowledgements

We would like to thank Emil Jeřábek for the proof of Lemma 4.3. We thank Andrei Bulatov for insightful discussions, and thank Magnus Wahlström for sharing his ideas and an initial version of the proof of Theorem 7.12. We thank the anonymous referees of Algorithmica for the suggestion that a balanced constraint language has a Maltsev embedding over a single integer ring \(\mathbb {Z}/q\mathbb {Z}\), and for comments that improved the presentation of the paper.

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Authors and Affiliations

  1. Birkbeck, University of London, Malet Street, Bloomsbury, London, WC1E 7HX, UK

    Hubie Chen

  2. Eindhoven University of Technology, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands

    Bart M. P. Jansen

  3. Institut für Informatik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099, Berlin, Germany

    Astrid Pieterse

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  1. Hubie Chen
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Correspondence to Astrid Pieterse.

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An extended abstract of this work, with the same title, appeared in the Proceedings of the 13th International Symposium on Parameterized and Exact Computation (IPEC 2018). This work was supported by NWO Gravitation grant “Networks”. This work was done while the third author was a PhD student at Eindhoven University of Technology.

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Chen, H., Jansen, B.M.P. & Pieterse, A. Best-Case and Worst-Case Sparsifiability of Boolean CSPs. Algorithmica 82, 2200–2242 (2020). https://doi.org/10.1007/s00453-019-00660-y

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