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  "body": "\n## Statement\n\nLet $\\mathbb{B}$ be a finite [[relational structure|MJF9M19E3E]] with finitely many relations.\nThe problem $\\mathrm{CSP}(\\mathbb{B})$ takes as input a finite structure $\\mathbb{A}$ of the same signature.\nIt asks whether there is a [[homomorphism|KBXSZC6BV1]] from $\\mathbb{A}$ to $\\mathbb{B}$.\nThe conjecture states that every such constraint satisfaction problem $\\mathrm{CSP}(\\mathbb{B})$ is either solvable in polynomial time or NP-complete.\n\n## Remarks\n\nThe conjecture is now a theorem: it follows from the [[CSP dichotomy theorem|ZA4YXJ9T1P]], proved independently by Andrei Bulatov and by Dmitriy Zhuk.\nIt is not a consequence of general complexity theory.\nBy Ladner's theorem, if $\\mathrm{P} \\ne \\mathrm{NP}$ then NP contains problems that are neither in P nor NP-complete.\nFeder and Vardi studied a logically defined fragment of NP called monotone monadic SNP, defined by three syntactic restrictions.\nThey showed that its problems are equivalent to fixed-template CSPs.\nThey also showed that dropping any one of the three restrictions gives a class that contains, up to polynomial-time equivalence, every problem in NP.\nSo, assuming $\\mathrm{P} \\ne \\mathrm{NP}$, a dichotomy can only hold for the restricted class, and CSPs are its natural representatives.\nBefore the proof, the conjecture was known for structures with a two-element domain (Schaefer), for undirected graphs (Hell and Nešetřil), and for three-element domains (Bulatov).\n\n## Explanations\n\n### L1 human\n\nFix a finite set of labels and a finite list of rules saying which combinations of labels are allowed.\nThe task is then: given a finite input, can its points be labelled so that every rule is obeyed?\nFeder and Vardi guessed that each such task is either easy, with a fast algorithm, or as hard as the hardest problems in NP.\nThere would be no task of intermediate difficulty in this family, even though such tasks exist elsewhere in NP if P differs from NP.\n",
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  "front": {
    "history": {
      "first_stated": "1993",
      "posed_by": "Tomás Feder and Moshe Y. Vardi"
    },
    "kind": "conjecture",
    "msc": [
      "68Q25",
      "08A70"
    ],
    "name": "Feder–Vardi dichotomy conjecture",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Constraint_satisfaction_problem"
      }
    ],
    "tag": "WNHZC07XVP",
    "uses": [
      "MJF9M19E3E",
      "KBXSZC6BV1"
    ]
  },
  "kind": "conjecture",
  "link": "https://tamnd.github.io/math2/node/WNHZC07XVP/",
  "links": [
    {
      "field": "uses",
      "target": "MJF9M19E3E"
    },
    {
      "field": "uses",
      "target": "KBXSZC6BV1"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/WNHZC07XVP.md",
  "msc": [
    "68Q25",
    "08A70"
  ],
  "name": "Feder–Vardi dichotomy conjecture",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/WNH/WNHZC07XVP.md",
  "status": null,
  "tag": "WNHZC07XVP",
  "uri": "m2://WNHZC07XVP",
  "url": "https://tamnd.github.io/math2/wiki/feder-vardi-dichotomy-conjecture/",
  "writing": {
    "tag": "WNHZC07XVP",
    "kind": "conjecture",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
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}