# The Power of the Combined Basic LP and Affine Relaxation for Promise CSPs

## TL;DR

This paper introduces a polynomial-time algorithm for promise CSPs that admit infinitely many symmetric polymorphisms, extending previous work and unifying solutions for Boolean CSPs by leveraging symmetry properties.

## Contribution

The authors develop a new algorithm that solves promise CSPs with symmetric polymorphisms, generalizing prior results and providing a complete characterization of its applicability.

## Key findings

- The algorithm solves all promise CSPs with infinitely many symmetric polymorphisms.
- It extends to block-symmetric polymorphisms, broadening its scope.
- Block symmetric polymorphisms are both sufficient and necessary for the algorithm's success.

## Abstract

In the field of constraint satisfaction problems (CSP), promise CSPs are an exciting new direction of study. In a promise CSP, each constraint comes in two forms: "strict" and "weak," and in the associated decision problem one must distinguish between being able to satisfy all the strict constraints versus not being able to satisfy all the weak constraints. The most commonly cited example of a promise CSP is the approximate graph coloring problem--which has recently seen exciting progress [BKO19, WZ20] benefiting from a systematic algebraic approach to promise CSPs based on "polymorphisms," operations that map tuples in the strict form of each constraint to tuples in the corresponding weak form. In this work, we present a simple algorithm which in polynomial time solves the decision problem for all promise CSPs that admit infinitely many symmetric polymorphisms, which are invariant under arbitrary coordinate permutations. This generalizes previous work of the first two authors [BG19]. We also extend this algorithm to a more general class of block-symmetric polymorphisms. As a corollary, this single algorithm solves all polynomial-time tractable Boolean CSPs simultaneously. These results give a new perspective on Schaefer's classic dichotomy theorem and shed further light on how symmetries of polymorphisms enable algorithms. Finally, we show that block symmetric polymorphisms are not only sufficient but also necessary for this algorithm to work, thus establishing its precise power