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  "body": "\n## Summary\n\nThe algebraic approach studies the constraint satisfaction problem of a fixed finite structure through the [[polymorphisms|A0K9EZRM7E]] of that structure.\nIts basic fact is that the complexity of the problem depends only on the clone of polymorphisms.\nPolymorphisms satisfying good identities, such as [[Taylor|QEAMRNRJJD]], [[Siggers|JYW1H3J8SX]] or cyclic identities, give efficient algorithms.\nThe approach led to the proof of the [[CSP dichotomy theorem|ZA4YXJ9T1P]]: every such problem is either in polynomial time or NP-complete.\n\n## Explanations\n\n### L1 human\n\nMany puzzles ask for values of variables that satisfy given constraints, for example colouring a map so that neighbours get different colours.\nIf the allowed constraints are fixed, one can ask how hard the puzzle is.\nThe algebraic approach looks for operations that combine several solutions into a new solution.\nFor example, for $2$-colouring a graph, the majority of three valid colourings is again a valid colouring, and $2$-colouring is easy.\nFor $3$-colouring no such good operation exists, and the problem is NP-complete.\n",
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