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Search-Space Reduction for Boolean MinCSPs via Essential Constraints

Authors Bart M. P. Jansen , Ruben F. A. Verhaegh

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ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Constraint and logic programming
Keywords
  • fixed-parameter tractability
  • constraint satisfaction problems

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Abstract

For a fixed set ℱ of Boolean constraint types, a MinCSP(ℱ)-instance consists of a formula F that applies m constraints from ℱ to a set of n Boolean variables. The goal is to remove a minimum subset of constraint applications from F to make the remaining formula satisfiable. Previous work characterized how the choice of ℱ affects its polynomial-time solvability and approximability. We extend a recently introduced preprocessing framework for graph problems to the problem above. Rephrased in the context of CSPs, this framework defines a constraint application from a given formula F as c-essential if it is contained in all c-approximate solutions to F. Being able to efficiently detect these essential parts of a solution reduces the search space of any follow-up FPT algorithms parameterized by the solution size and yields an immediate asymptotic improvement to the runtime of such algorithms. In this work, we present a dichotomy theorem that distinguishes constraint sets ℱ for which c_ℱ-essential constraint applications can be detected efficiently for some c_{ℱ} ∈ 𝒪(1), from those for which this task is intractable under established complexity-theoretic conjectures. Our results show that for any set ℱ of bijunctive constraints, there is a polynomial-time algorithm that detects 𝒪(1)-essential constraint applications. This contrasts the fact that constant-factor approximating a bijunctive MinCSP(ℱ)-problem is intractable under the Unique Games Conjecture.

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Bart M. P. Jansen and Ruben F. A. Verhaegh. Search-Space Reduction for Boolean MinCSPs via Essential Constraints. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 22:1-22:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SWAT.2026.22

Author Details

Bart M. P. Jansen
  • Eindhoven University of Technology, The Netherlands
Ruben F. A. Verhaegh
  • Eindhoven University of Technology, The Netherlands

References

  1. Tobias Achterberg, Robert E. Bixby, Zonghao Gu, Edward Rothberg, and Dieter Weninger. Presolve reductions in mixed integer programming. Technical Report 16-44, ZIB, Takustr.7, 14195 Berlin, 2016. URL: http://nbn-resolving.de/urn:nbn:de:0297-zib-60370.
  2. Sanjeev Arora and Boaz Barak. Computational Complexity - A Modern Approach. Cambridge University Press, 2009. URL: https://doi.org/10.1017/CBO9780511804090.
  3. Bengt Aspvall, Michael F. Plass, and Robert Endre Tarjan. A linear-time algorithm for testing the truth of certain quantified boolean formulas. Inf. Process. Lett., 8(3):121-123, 1979. URL: https://doi.org/10.1016/0020-0190(79)90002-4.
  4. Vijay Bhattiprolu, Venkatesan Guruswami, and Xuandi Ren. PCP-free APX-hardness of nearest codeword and minimum distance. Electron. Colloquium Comput. Complex., TR25, 2025. URL: https://eccc.weizmann.ac.il/report/2025/029.
  5. Édouard Bonnet, László Egri, and Dániel Marx. Fixed-parameter approximability of Boolean MinCSPs. In Piotr Sankowski and Christos D. Zaroliagis, editors, 24th Annual European Symposium on Algorithms, ESA 2016, Aarhus, Denmark, August 22-24, 2016, volume 57 of LIPIcs, pages 18:1-18:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL: https://doi.org/10.4230/LIPIcs.ESA.2016.18.
  6. Benjamin Merlin Bumpus, Bart M. P. Jansen, and Jari J. H. de Kroon. Search-space reduction via essential vertices. SIAM J. Discret. Math., 38(3):2392-2415, 2024. URL: https://doi.org/10.1137/23M1589347.
  7. Fedor Fomin, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. Kernelization: theory of parameterized preprocessing. Cambridge University Press, 2019. Google Scholar
  8. L. R. Ford and D. R. Fulkerson. Maximal flow through a network. Canadian Journal of Mathematics, 8:399-404, 1956. URL: https://doi.org/10.4153/CJM-1956-045-5.
  9. Yoichi Iwata, Magnus Wahlström, and Yuichi Yoshida. Half-integrality, lp-branching, and FPT algorithms. SIAM J. Comput., 45(4):1377-1411, 2016. URL: https://doi.org/10.1137/140962838.
  10. Yoichi Iwata, Yutaro Yamaguchi, and Yuichi Yoshida. 0/1/All CSPs, half-integral A-path packing, and linear-time FPT algorithms. In Mikkel Thorup, editor, 59th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2018, Paris, France, October 7-9, 2018, pages 462-473. IEEE Computer Society, 2018. URL: https://doi.org/10.1109/FOCS.2018.00051.
  11. Bart M. P. Jansen and Ruben F. A. Verhaegh. Search-space reduction via essential vertices revisited: Vertex multicut and cograph deletion. J. Comput. Syst. Sci., 156:103730, 2026. URL: https://doi.org/10.1016/J.JCSS.2025.103730.
  12. Bart M. P. Jansen and Michal Wlodarczyk. Optimal polynomial-time compression for Boolean max CSP. ACM Trans. Comput. Theory, 16(1):4:1-4:20, 2024. URL: https://doi.org/10.1145/3624704.
  13. Sanjeev Khanna, Madhu Sudan, Luca Trevisan, and David P. Williamson. The approximability of constraint satisfaction problems. SIAM J. Comput., 30(6):1863-1920, 2000. URL: https://doi.org/10.1137/S0097539799349948.
  14. Subhash Khot. On the power of unique 2-prover 1-round games. In John H. Reif, editor, Proceedings on 34th Annual ACM Symposium on Theory of Computing, May 19-21, 2002, Montréal, Québec, Canada, pages 767-775. ACM, 2002. URL: https://doi.org/10.1145/509907.510017.
  15. Eun Jung Kim, Stefan Kratsch, Marcin Pilipczuk, and Magnus Wahlström. Flow-augmentation III: Complexity dichotomy for Boolean CSPs parameterized by the number of unsatisfied constraints. SIAM J. Comput., 54(4):1065-1137, 2025. URL: https://doi.org/10.1137/23M1553698.
  16. Stefan Kratsch, Dániel Marx, and Magnus Wahlström. Parameterized complexity and kernelizability of max ones and exact ones problems. ACM Trans. Comput. Theory, 8(1):1:1-1:28, 2016. URL: https://doi.org/10.1145/2858787.
  17. Stefan Kratsch and Magnus Wahlström. Preprocessing of min ones problems: A dichotomy. In Samson Abramsky, Cyril Gavoille, Claude Kirchner, Friedhelm Meyer auf der Heide, and Paul G. Spirakis, editors, Automata, Languages and Programming, 37th International Colloquium, ICALP 2010, Bordeaux, France, July 6-10, 2010, Proceedings, Part I, volume 6198 of Lecture Notes in Computer Science, pages 653-665. Springer, 2010. URL: https://doi.org/10.1007/978-3-642-14165-2_55.
  18. Stefan Kratsch and Magnus Wahlström. Representative sets and irrelevant vertices: New tools for kernelization. J. ACM, 67(3):16:1-16:50, 2020. URL: https://doi.org/10.1145/3390887.
  19. Andrei Krokhin and Stanislav Zivny. The Constraint Satisfaction Problem: Complexity and Approximability, volume 7 of Dagstuhl Follow-Ups. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, Dagstuhl, Germany, 2017. URL: https://doi.org/10.4230/DFU.Vol7.15301.
  20. M. S. Ramanujan and Saket Saurabh. Linear-time parameterized algorithms via skew-symmetric multicuts. ACM Trans. Algorithms, 13(4):46:1-46:25, 2017. URL: https://doi.org/10.1145/3128600.
  21. Thomas J. Schaefer. The complexity of satisfiability problems. In Richard J. Lipton, Walter A. Burkhard, Walter J. Savitch, Emily P. Friedman, and Alfred V. Aho, editors, Proceedings of the 10th Annual ACM Symposium on Theory of Computing, May 1-3, 1978, San Diego, California, USA, pages 216-226. ACM, 1978. URL: https://doi.org/10.1145/800133.804350.
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