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  "body": "\n## Definition\n\nLet $\\mathbb{A} = (A; R_1, \\dots, R_k)$ be a relational structure with a finite nonempty universe $A$ and finitely many relations.\nThe *constraint satisfaction problem* $\\operatorname{CSP}(\\mathbb{A})$ is the following decision problem.\nThe input is a finite relational structure $\\mathbb{X}$ with relations of the same arities.\nThe question is whether there is a homomorphism from $\\mathbb{X}$ to $\\mathbb{A}$, a map preserving each relation.\nThe *polymorphism clone* of $\\mathbb{A}$ is the [[clone|PTKR703G6A]] $\\operatorname{Pol}(\\mathbb{A})$ of all operations on $A$ that are [[polymorphisms|A0K9EZRM7E]] of every $R_i$.\nThe *algebraic approach* is the study of the computational complexity of $\\operatorname{CSP}(\\mathbb{A})$ through the algebra $(A; \\operatorname{Pol}(\\mathbb{A}))$.\n\n## Remarks\n\nThe approach rests on a reduction principle.\nIf $\\mathbb{A}$ and $\\mathbb{B}$ share the universe $A$ and $\\operatorname{Pol}(\\mathbb{A}) \\subseteq \\operatorname{Pol}(\\mathbb{B})$, then $\\operatorname{CSP}(\\mathbb{B})$ reduces to $\\operatorname{CSP}(\\mathbb{A})$ in polynomial time.\nThe reason is that each relation of $\\mathbb{B}$ is then primitive positive definable from $\\mathbb{A}$, by the [[Pol–Inv Galois connection|BQ0AC9QX7E]].\nExample: for the complete graph $K_2$, $\\operatorname{CSP}(K_2)$ asks whether a graph is $2$-colourable, as a [[graph homomorphism|052JJ9YVNM]] to $K_2$.\nThe majority operation on $\\set{0, 1}$ is a polymorphism of the edge relation $\\set{(0, 1), (1, 0)}$, and the problem is in polynomial time.\nNon-example of tractability: every polymorphism of $K_3$ has the form $\\sigma(x_i)$ for a permutation $\\sigma$ of $\\set{0, 1, 2}$.\nSo $K_3$ has no Taylor polymorphism, and $3$-colouring is NP-complete.\n",
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