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Optimal Inapproximability of Generalized Linear Equations over a Finite Group

Authors Amey Bhangale , Yezhou Zhang

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Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
Keywords
  • Constraint satisfaction problems
  • inapproximability
  • approximation algorithms
  • non-abelian groups
  • Fourier analysis

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Abstract

Constraint satisfaction problems (CSPs) consist of a set of variables taking values from some finite domain and a set of local constraints on these variables. The objective is to find an assignment to the variables that maximizes the fraction of satisfied constraints.
In this work, we study the CSP where the constraints are generalized linear equations over a finite group G. More specifically, for a given S ⊆ G, the constraints in this CSP are of the form addition of the values to the variables (similarly, product for non-abelian groups) belongs to the set S. We give an approximation algorithm for this problem on satisfiable instances and show that it is optimal for certain S assuming 𝐏≠ NP. 
This natural predicate is one of the very few known predicates that are approximation resistant on almost satisfiable instances, assuming 𝐏≠ NP, but admits a non-trivial approximation algorithm on satisfiable instances.

Cite As Get BibTex

Amey Bhangale and Yezhou Zhang. Optimal Inapproximability of Generalized Linear Equations over a Finite Group. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 30:1-30:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.30

Author Details

Amey Bhangale
  • Department of Computer Science and Engineering, University of California, Riverside, CA, USA
Yezhou Zhang
  • Department of Computer Science and Engineering, University of California, Riverside, CA, USA

Funding

  • Bhangale, Amey: Supported by the Hellman Fellowship award and NSF CAREER award 2440882.
  • Zhang, Yezhou: Supported by the Hellman Fellowship award and NSF CAREER award 2440882.

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