Abelian embeddings make useful promises: approximation of detrivialised equations

Abstract

We study the following family of search problems: given a system of equations $x+y+z\equiv 0\pmod p$ promised to admit a solution that is nonconstant on every equation, find a $c$-colouring of the variables that makes no equation monochromatic. For $p\in\{2,3\}$, the problem recovers known tractable Promise Constraint Satisfaction Problems (PCSPs). For $p > 3$, we obtain a complete complexity classification: the problem is polynomial-time solvable for $c \geq 4$, yet even its decision version is NP-hard for $c = 3$, and it is not well-defined for $c = 2$. Our main conceptual contribution is a general tractability framework for PCSPs based on Abelian embeddings, a notion introduced by Bhangale, Khot, and Minzer [STOC 2022] in the study of quantitative approximation of satisfiable CSPs. We show that suitable Abelian embeddings yield polynomial-time PCSP reductions realised by singleton algorithms in the sense of Zhuk [FOCS 2026]. On the hardness side, the promise relation is far removed from the reconfigurable regime covered by the conditional hardness framework of Banakh, Barsukov, and Nakajima [LICS 2026]; our proof instead relies on a new additive-combinatorial rigidity theorem classifying special $3$-colourings of $\mathbb{Z}_p$.