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What makes a computational problem easy (e.g., in P, that is, solvable inpolynomial time) or hard (e.g., NP-hard)? This fundamental question now has asatisfactory answer for a quite broad class of computational problems, socalled fixed-template constraint satisfaction problems (CSPs) -- it has turnedout that their complexity is captured by a certain specific form of symmetry.This paper explains an extension of this theory to a much broader class ofcomputational problems, the promise CSPs, which includes relaxed versions ofCSPs such as the problem of finding a 137-coloring of a 3-colorable graph.

Algebraic Theory of Promise Constraint Satisfaction Problems, First Steps