Triviality of promise polymorphisms

2026-07-29

AI Summary

This work addresses the problem of determining the triviality of polymorphisms in promise constraint satisfaction problems (PCSPs) with two predicates. Extending the existing framework for single-predicate cases to the more general setting where a promise constraint $P \subseteq Q$ involves two predicates, the paper introduces a relaxed notion of certificates. By leveraging tools from universal algebra—specifically polymorphism theory—combined with combinatorial structure analysis and classification of obstacle sets, the authors establish that global triviality can be decided by examining only unary or binary polymorphisms. This result substantially reduces the computational complexity of the triviality test and provides an efficient criterion for analyzing the tractability of promise CSPs.

Abstract

Given two $m$-ary predicates $P,Q$, an $n$-ary polymorphism is a tuple $(f_1,\dots,f_m)$ of functions such that $x^{(1)},\dots,x^{(n)} \in P$ implies $(f_1(y_1),\dots,f_m(y_m)) \in Q$, where $y_i = (x^{(1)}_i,\dots,x^{(m)}_i)$. This generalizes the usual definition in universal algebra, in which $P = Q$ and $f_1 = \cdots = f_m$. In earlier work, we studied when all polymorphisms of a single predicate are "trivial": either all depend on a single coordinate (common to all of them), or they constitute a "certificate" for the predicate. We showed that it suffices to check this condition for $2$-ary polymorphisms, and even for $1$-ary polymorphisms, modulo an explicit list of obstructions. In this paper we generalize the first result to the $P,Q$ setting, for a relaxed notion of certificate. We also generalize the second result in the promise setting, in which $P,Q$ range over the same alphabets and $P \subseteq Q$.

Problem

• promise polymorphisms
• triviality
• predicates
• universal algebra
• certificate

Innovation

• promise polymorphisms
• triviality
• constraint satisfaction problems
• universal algebra
• certificate