Translationally Invariant Constraint Optimization Problems

Authors Dorit Aharonov, Sandy Irani

Thumbnail PDF


  • Filesize: 0.74 MB
  • 15 pages

Document Identifiers

Author Details

Dorit Aharonov
  • Department of Computer Science and Engineering, Hebrew University, Jerusalem, Israel
Sandy Irani
  • Department of Computer Science, University of California Irvine, CA, USA
  • The Simons Institute for the Theory of Computing, University of California Berkeley, CA, USA


We are grateful to the Simons Institute for the Theory of Computing, at whose program on the "The Quantum Wave in Computing" this collaboration began.

Cite AsGet BibTex

Dorit Aharonov and Sandy Irani. Translationally Invariant Constraint Optimization Problems. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 23:1-23:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


We study the complexity of classical constraint satisfaction problems on a 2D grid. Specifically, we consider the computational complexity of function versions of such problems, with the additional restriction that the constraints are translationally invariant, namely, the variables are located at the vertices of a 2D grid and the constraint between every pair of adjacent variables is the same in each dimension. The only input to the problem is thus the size of the grid. This problem is equivalent to one of the most interesting problems in classical physics, namely, computing the lowest energy of a classical system of particles on the grid. We provide a tight characterization of the complexity of this problem, and show that it is complete for the class FP^NEXP. Gottesman and Irani (FOCS 2009) also studied classical constraint satisfaction problems using this strong notion of translational-invariance; they show that the problem of deciding whether the cost of the optimal assignment is below a given threshold is NEXP-complete. Our result is thus a strengthening of their result from the decision version to the function version of the problem. Our result can also be viewed as a generalization to the translationally invariant setting, of Krentel’s famous result from 1988, showing that the function version of SAT is complete for the class FP^NP. An essential ingredient in the proof is a study of the computational complexity of a gapped variant of the problem. We show that it is NEXP-hard to approximate the cost of the optimal assignment to within an additive error of Ω(N^(1/4)), where the grid size is N × N. To the best of our knowledge, no gapped result is known for CSPs on the grid, even in the non-translationally invariant case. This might be of independent interest. As a byproduct of our results, we also show that a decision version of the optimization problem which asks whether the cost of the optimal assignment is odd or even is also complete for P^NEXP.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Constraint satisfaction
  • Tiling
  • Translational-invariance


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


  1. Ian Affleck, Tom Kennedy, Elliott H. Lieb, and Hal Tasaki. Rigorous results on valence-bond ground states in antiferromagnets. Phys. Rev. Lett., 59:799-802, August 1987. URL:
  2. Dorit Aharonov, Daniel Gottesman, Sandy Irani, and Julia Kempe. The power of quantum systems on a line. Communications in Mathematical Physics, 287(1):41-65, January 2009. URL:
  3. Dorit Aharonov and Sandy Irani. Hamiltonian complexity in the thermodynamic limit. In Stefano Leonardi and Anupam Gupta, editors, STOC '22: 54th Annual ACM SIGACT Symposium on Theory of Computing, Rome, Italy, June 20 - 24, 2022, pages 750-763. ACM, 2022. URL:
  4. Dorit Aharonov and Sandy Irani. Translationally invariant constraint optimization problems, 2022. URL:
  5. Andris Ambainis. On physical problems that are slightly more difficult than qma. In 2014 IEEE 29th Conference on Computational Complexity (CCC), pages 32-43, 2014. URL:
  6. F Baharona. On the computational complexity of ising spin glass models. Journal of Physics A: Mathematical and General, 15(10):3241-3253, 1982. Google Scholar
  7. Johannes Bausch, Toby Cubitt, and Maris Ozols. The complexity of translationally invariant spin chains with low local dimension. Annales Henri Poincaré, 18(11):3449-3513, October 2017. URL:
  8. Toby S. Cubitt, David Perez-Garcia, and Michael M. Wolf. Undecidability of the spectral gap. Nature, 528(7581):207-211, December 2015. URL:
  9. Sevag Gharibian, Yichen Huang, Zeph Landau, and Seung Woo Shin. Quantum hamiltonian complexity. Foundations and Trends® in Theoretical Computer Science, 10(3):159-282, 2015. URL:
  10. Sevag Gharibian, Stephen Piddock, and Justin Yirka. Oracle complexity classes and local measurements on physical hamiltonians. arXiv, 2019. URL:
  11. Sevag Gharibian and Justin Yirka. The complexity of simulating local measurements on quantum systems. Quantum, 3:189, September 2019. URL:
  12. Daniel Gottesman and Sandy Irani. The quantum and classical complexity of translationally invariant tiling and hamiltonian problems. Theory of Computing, 9(2):31-116, 2013. URL:
  13. Sorin Istrail. Statistical mechanics, three-dimensionality and np-completeness. i. universality of intractability for the partition function of the ising model across non-planar lattices. In Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, May 21-23, 2000, Portland, OR, USA, pages 87-96, January 2000. URL:
  14. A. Yu. Kitaev, A. H. Shen, and M. N. Vyalyi. Classical and Quantum Computation. American Mathematical Society, USA, 2002. Google Scholar
  15. Mark W. Krentel. The complexity of optimization problems. In Alan L. Selman, editor, Structure in Complexity Theory, pages 218-218, Berlin, Heidelberg, 1986. Springer Berlin Heidelberg. Google Scholar
  16. R. Oliveira and B. Terhal. The complexity of quantum spin systems on a two-dimensional square lattice. arXiv, 2005. URL:
  17. Christos H. Papadimitriou. Computational complexity. Addison-Wesley, 1994. Google Scholar
  18. Raphael Robinson. Undecidability and nonperiodicity for the tilings of the plane. Invent. Math., 12:177-209, 1971. Google Scholar
  19. Hao Wang. Proving theorems by pattern recognition. Communications of the ACM, 3(4):220-234, 1960. Google Scholar
  20. James D. Watson, Johannes Bausch, and Sevag Gharibian. The complexity of translationally invariant problems beyond ground state energies. arXiv, 2020. URL:
  21. James D. Watson and Toby S. Cubitt. Computational complexity of the ground state energy density problem. arXiv, 2021. 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