Translationally Invariant Constraint Optimization Problems

Dorit Aharonov*, Sandy Irani*

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


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 FPNEXP. 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 FPNP. 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(N1/4), 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.

Original languageAmerican English
Title of host publication38th Computational Complexity Conference, CCC 2023
EditorsAmnon Ta-Shma
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Number of pages15
ISBN (Electronic)9783959772822
StatePublished - Jul 2023
Event38th Computational Complexity Conference, CCC 2023 - Warwick, United Kingdom
Duration: 17 Jul 202320 Jul 2023

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
ISSN (Print)1868-8969


Conference38th Computational Complexity Conference, CCC 2023
Country/TerritoryUnited Kingdom

Bibliographical note

Publisher Copyright:
© Dorit Aharonov and Sandy Irani; licensed under Creative Commons License CC-BY 4.0.


  • Constraint satisfaction
  • Tiling
  • Translational-invariance


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