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On the Constant-Factor Approximability of Minimum Cost Constraint Satisfaction Problems

Authors Ian DeHaan , Neng Huang , Euiwoong Lee

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ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Constraint satisfaction problems
  • approximation algorithms
  • polymorphisms

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Abstract

We study minimum cost constraint satisfaction problems (MinCostCSP) through the algebraic lens. We show that for any constraint language Γ which has the dual discriminator operation as a polymorphism, there exists a |D|-approximation algorithm for MinCostCSP(Γ) where D is the domain. Complementing our algorithmic result, we show that any constraint language Γ where MinCostCSP(Γ) admits a constant-factor approximation must have a near-unanimity (NU) polymorphism unless P = NP, extending a similar result by Dalmau et al. on MinCSPs. These results imply a dichotomy of constant-factor approximability for constraint languages that contain all permutation relations (a natural generalization for Boolean CSPs that allow variable negation): either MinCostCSP(Γ) has an NU polymorphism and is |D|-approximable, or it does not have any NU polymorphism and is NP-hard to approximate within any constant factor. Finally, we present a constraint language which has a majority polymorphism, but is nonetheless NP-hard to approximate within any constant factor assuming the Unique Games Conjecture, showing that the condition of having an NU polymorphism is in general not sufficient unless UGC fails.

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Ian DeHaan, Neng Huang, and Euiwoong Lee. On the Constant-Factor Approximability of Minimum Cost Constraint Satisfaction Problems. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 19:1-19:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2025.19

Author Details

Ian DeHaan
  • University of Michigan, Ann Arbor, MI, USA
Neng Huang
  • University of Michigan, Ann Arbor, MI, USA
Euiwoong Lee
  • University of Michigan, Ann Arbor, MI, USA

Funding

  • Lee, Euiwoong: Supported in part by NSF grant CCF-2236669.

Acknowledgements

The authors would like to thank anonymous reviewers for their helpful comments.

References

  1. Kirby A Baker and Alden F Pixley. Polynomial interpolation and the chinese remainder theorem for algebraic systems. Mathematische Zeitschrift, 143:165-174, 1975. Google Scholar
  2. Nikhil Bansal and Subhash Khot. Inapproximability of hypergraph vertex cover and applications to scheduling problems. In International Colloquium on Automata, Languages, and Programming, pages 250-261. Springer, 2010. URL: https://doi.org/10.1007/978-3-642-14165-2_22.
  3. Libor Barto. The dichotomy for conservative constraint satisfaction problems revisited. In 2011 IEEE 26th Annual Symposium on Logic in Computer Science, pages 301-310. IEEE, 2011. URL: https://doi.org/10.1109/LICS.2011.25.
  4. Libor Barto. The collapse of the bounded width hierarchy. Journal of Logic and Computation, 26(3):923-943, 2014. Google Scholar
  5. Libor Barto and Marcin Kozik. Constraint satisfaction problems solvable by local consistency methods. Journal of the ACM (JACM), 61(1):1-19, 2014. URL: https://doi.org/10.1145/2556646.
  6. Libor Barto, Andrei Krokhin, and Ross Willard. Polymorphisms, and How to Use Them. In Andrei Krokhin and Stanislav Zivny, editors, The Constraint Satisfaction Problem: Complexity and Approximability, volume 7 of Dagstuhl Follow-Ups, pages 1-44. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, Dagstuhl, Germany, 2017. URL: https://doi.org/10.4230/DFU.Vol7.15301.1.
  7. VG Bondarchuk, LA Kaluzhnin, VN Kotov, and BA Romov. Galois theory for post algebras. i-ii. Kibernetika, 3:1-10, 1969. Google Scholar
  8. Joshua Brakensiek, Neng Huang, Aaron Potechin, and Uri Zwick. On the mysteries of max nae-sat. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 484-503. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976465.30.
  9. Joshua Brakensiek, Neng Huang, Aaron Potechin, and Uri Zwick. Separating max 2-and, max di-cut and max cut. In 2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS), pages 234-252. IEEE, 2023. URL: https://doi.org/10.1109/FOCS57990.2023.00023.
  10. Andrei A. Bulatov. Tractable conservative constraint satisfaction problems. In 18th Annual IEEE Symposium of Logic in Computer Science, 2003. Proceedings., pages 321-330. IEEE, 2003. Google Scholar
  11. Andrei A. Bulatov. Conservative constraint satisfaction re-revisited. Journal of Computer and System Sciences, 82(2):347-356, 2016. URL: https://doi.org/10.1016/J.JCSS.2015.07.004.
  12. Andrei A Bulatov. A dichotomy theorem for nonuniform csps. In 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS), pages 319-330. IEEE, 2017. URL: https://doi.org/10.1109/FOCS.2017.37.
  13. Qi Cheng and Daqing Wan. A deterministic reduction for the gap minimum distance problem. IEEE Transactions on Information Theory, 58(11):6935-6941, 2012. URL: https://doi.org/10.1109/TIT.2012.2209198.
  14. Martin C. Cooper, David A. Cohen, and Peter G. Jeavons. Characterising tractable constraints. Artificial Intelligence, 65(2):347-361, 1994. URL: https://doi.org/10.1016/0004-3702(94)90021-3.
  15. Victor Dalmau. A new tractable class of constraint satisfaction problems. Annals of Mathematics and Artificial Intelligence, 44:61-85, 2005. URL: https://doi.org/10.1007/S10472-005-1810-9.
  16. Víctor Dalmau, Marcin Kozik, Andrei Krokhin, Konstantin Makarychev, Yury Makarychev, and Jakub Oprsal. Robust algorithms with polynomial loss for near-unanimity csps. SIAM Journal on Computing, 48(6):1763-1795, 2019. URL: https://doi.org/10.1137/18M1163932.
  17. Víctor Dalmau and Andrei Krokhin. Robust satisfiability for csps: Hardness and algorithmic results. ACM Transactions on Computation Theory (TOCT), 5(4):1-25, 2013. URL: https://doi.org/10.1145/2540090.
  18. Víctor Dalmau, Andrei Krokhin, and Rajsekar Manokaran. Towards a characterization of constant-factor approximable finite-valued csps. Journal of Computer and System Sciences, 97:14-27, 2018. URL: https://doi.org/10.1016/J.JCSS.2018.03.003.
  19. Irit Dinur, Venkatesan Guruswami, Subhash Khot, and Oded Regev. A new multilayered pcp and the hardness of hypergraph vertex cover. SIAM Journal on Computing, 34(5):1129-1146, 2005. URL: https://doi.org/10.1137/S0097539704443057.
  20. I Dumer, D. Micciancio, and M. Sudan. Hardness of approximating the minimum distance of a linear code. IEEE Transactions on Information Theory, 49(1):22-37, 2003. URL: https://doi.org/10.1109/TIT.2002.806118.
  21. Alina Ene, Jan Vondrak, and Yi Wu. Local distribution and the symmetry gap: Approximability of multiway partitioning problems. arXiv preprint, 2015. URL: https://arxiv.org/abs/1503.03905.
  22. Tomás Feder and Moshe Y Vardi. The computational structure of monotone monadic snp and constraint satisfaction: A study through datalog and group theory. SIAM Journal on Computing, 28(1):57-104, 1998. URL: https://doi.org/10.1137/S0097539794266766.
  23. David Geiger. Closed systems of functions and predicates. Pacific journal of mathematics, 27(1):95-100, 1968. Google Scholar
  24. Dan Gusfield and Leonard Pitt. A bounded approximation for the minimum cost 2-sat problem. Algorithmica, 8(1):103-117, 1992. URL: https://doi.org/10.1007/BF01758838.
  25. Gregory Gutin, Pavol Hell, Arash Rafiey, and Anders Yeo. A dichotomy for minimum cost graph homomorphisms. European Journal of Combinatorics, 29(4):900-911, 2008. URL: https://doi.org/10.1016/J.EJC.2007.11.012.
  26. Pavol Hell, Monaldo Mastrolilli, Mayssam Mohammadi Nevisi, and Arash Rafiey. Approximation of minimum cost homomorphisms. In Algorithms-ESA 2012: 20th Annual European Symposium, Ljubljana, Slovenia, September 10-12, 2012. Proceedings 20, pages 587-598. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-33090-2_51.
  27. Pavol Hell and Arash Rafiey. The dichotomy of minimum cost homomorphism problems for digraphs. SIAM Journal on Discrete Mathematics, 26(4):1597-1608, 2012. URL: https://doi.org/10.1137/100783856.
  28. Peter Jeavons, David Cohen, and Martin C Cooper. Constraints, consistency and closure. Artificial Intelligence, 101(1-2):251-265, 1998. URL: https://doi.org/10.1016/S0004-3702(98)00022-8.
  29. Peter Jonsson and Gustav Nordh. Introduction to the maximum solution problem. Complexity of Constraints: An Overview of Current Research Themes, pages 255-282, 2008. URL: https://doi.org/10.1007/978-3-540-92800-3_10.
  30. Sanjeev Khanna, Madhu Sudan, Luca Trevisan, and David P Williamson. The approximability of constraint satisfaction problems. SIAM Journal on Computing, 30(6):1863-1920, 2001. URL: https://doi.org/10.1137/S0097539799349948.
  31. Subhash Khot. On the power of unique 2-prover 1-round games. In Proceedings of the thiry-fourth annual ACM symposium on Theory of computing, pages 767-775, 2002. URL: https://doi.org/10.1145/509907.510017.
  32. Subhash Khot, Guy Kindler, Elchanan Mossel, and Ryan O’Donnell. Optimal inapproximability results for MAX-CUT and other 2-variable CSPs? SIAM Journal on Computing, 37(1):319-357, 2007. URL: https://doi.org/10.1137/S0097539705447372.
  33. Subhash Khot and Oded Regev. Vertex cover might be hard to approximate to within 2- ε. Journal of Computer and System Sciences, 74(3):335-349, 2008. Google Scholar
  34. Andrei Krokhin and Stanislav Zivny. The Complexity of Valued CSPs. In Andrei Krokhin and Stanislav Zivny, editors, The Constraint Satisfaction Problem: Complexity and Approximability, volume 7 of Dagstuhl Follow-Ups, pages 233-266. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, Dagstuhl, Germany, 2017. URL: https://doi.org/10.4230/DFU.Vol7.15301.233.
  35. Amit Kumar, Rajsekar Manokaran, Madhur Tulsiani, and Nisheeth K Vishnoi. On lp-based approximability for strict csps. In Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1560-1573. SIAM, 2011. URL: https://doi.org/10.1137/1.9781611973082.121.
  36. S.S. Marchenkov. Homogeneous algebras. Problemy Kibernetiki, 39:85-106, 1982. Google Scholar
  37. E. Marczewski. Homogeneous algebras and homogeneous operations. Fund. Math, 56(8):103, 1964. Google Scholar
  38. E.L. Post. The two-valued iterative systems of mathematical logic. Annals of Mathematics Studies, 1941. Google Scholar
  39. Akbar Rafiey, Arash Rafiey, and Thiago Santos. Toward a dichotomy for approximation of H-coloring. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019), pages 91:1-91:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.ICALP.2019.91.
  40. Prasad Raghavendra. Optimal algorithms and inapproximability results for every csp? In Proceedings of the fortieth annual ACM symposium on Theory of computing, pages 245-254, 2008. URL: https://doi.org/10.1145/1374376.1374414.
  41. Thomas J Schaefer. The complexity of satisfiability problems. In Proceedings of the tenth annual ACM symposium on Theory of computing, pages 216-226, 1978. URL: https://doi.org/10.1145/800133.804350.
  42. Hanif D Sherali and Warren P Adams. A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM Journal on Discrete Mathematics, 3(3):411-430, 1990. URL: https://doi.org/10.1137/0403036.
  43. Ágnes Szendrei. Clones in universal algebra. Presses de l'Université de Montréal, 1986. Google Scholar
  44. Rustem Takhanov. A dichotomy theorem for the general minimum cost homomorphism problem. In 27th International Symposium on Theoretical Aspects of Computer Science (2010), pages 657-668. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2010. URL: https://doi.org/10.4230/LIPICS.STACS.2010.2493.
  45. Johan Thapper and Stanislav Živnỳ. The complexity of finite-valued csps. Journal of the ACM (JACM), 63(4):1-33, 2016. URL: https://doi.org/10.1145/2974019.
  46. Dmitriy Zhuk. A proof of the csp dichotomy conjecture. Journal of the ACM (JACM), 67(5):1-78, 2020. URL: https://doi.org/10.1145/3402029.
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