{
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  "slug": "snp-complexity",
  "title": "SNP (complexity)",
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  "lead": "In computational complexity theory, SNP (from Strict NP) is a complexity class containing a limited subset of NP based on its logical characterization in terms of graph-theoretical properties. It forms the basis for the definition of the class MaxSNP of optimization problems. It is defined as the class of problems that are properties of relational structures (such as graphs) expressible by a second-order logic formula of the following form: : \\exists S_1 \\dots \\exists S_\\ell \\, \\forall v_1 \\dots \\forall v_m \\,\\phi(R_1,\\dots,R_k,S_1,\\dots,S_\\ell,v_1,\\dots,v_m) where R_1,\\dots,R_k are relations of the structure (such as the adjacency relation, for a graph), S_1,\\dots,S_\\ell are unknown relations (sets of tuples of vertices), and \\phi is a quantifier-free formula: any boolean combination of the relations. That is, only existential second-order quantification (over relations) is allowed and only universal first-order quantification (over vertices) is allowed. If existential quantification over vertices were also allowed, the resulting complexity class would be equal to NP (more precisely, the class of those properties of relational structures that are in NP), a fact known as Fagin's theorem.…",
  "body": "In computational complexity theory, SNP (from Strict NP) is a complexity class containing a limited subset of NP based on its logical characterization in terms of graph-theoretical properties. It forms the basis for the definition of the class MaxSNP of optimization problems. It is defined as the class of problems that are properties of relational structures (such as graphs) expressible by a second-order logic formula of the following form: \\exists S_1 \\dots \\exists S_\\ell \\, \\forall v_1 \\dots \\forall v_m \\,\\phi(R_1,\\dots,R_k,S_1,\\dots,S_\\ell,v_1,\\dots,v_m) where R_1,\\dots,R_k are relations of the structure (such as the adjacency relation, for a graph), S_1,\\dots,S_\\ell are unknown relations (sets of tuples of vertices), and \\phi is a quantifier-free formula: any boolean combination of the relations. That is, only existential second-order quantification (over relations) is allowed and only universal first-order quantification (over vertices) is allowed. If existential quantification over vertices were also allowed, the resulting complexity class would be equal to NP (more precisely, the class of those properties of relational structures that are in NP), a fact known as Fagin's theorem. For example, SNP contains 3-Coloring (the problem of determining whether a given graph is 3-colorable), because it can be expressed by the following formula: \\exists S_1 \\exists S_2 \\exists S_3 \\, \\forall u \\forall v \\, \\bigl( S_1(u) \\vee S_2(u) \\vee S_3(u) \\bigr) \\, \\wedge \\, \\bigl( E(u,v)\\,\\implies\\,(\\neg S_1(u) \\vee \\neg S_1(v))\\,\\wedge\\,\\left(\\neg S_2(u) \\vee \\neg S_2(v)\\right)\\,\\wedge\\,(\\neg S_3(u) \\vee \\neg S_3(v)) \\bigr) Here E denotes the adjacency relation of the input graph, while the sets (unary relations) S_1,S_2,S_3 correspond to sets of vertices colored with one of the 3 colors. Similarly, SNP contains the k-SAT problem: the boolean satisfiability problem (SAT) where the formula is restricted to conjunctive normal form and to at most k literals per clause, where k is fixed. In particular, every MaxSNP-complete problem (under L-reductions or under AP-reductions) is also APX-complete (under PTAS reductions), and hence does not admit a PTAS unless P=NP. However, the proof of this relies on the PCP theorem, while proofs of MaxSNP-completeness are often elementary. ==External links==",
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    "id": "enwiki-001298950027",
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    "title": "SNP (complexity)",
    "author": [
      {
        "family": "Anticloud FZ LLE",
        "literal": "Anticloud FZ LLE"
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    "container-title": "encyclopegia.org",
    "publisher": "Anticloud FZ LLE",
    "publisher-place": "Dubai, United Arab Emirates",
    "URL": "https://encyclopegia.org/a/snp-complexity",
    "issued": {
      "date-parts": [
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          2026,
          11,
          15
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    "DOI": "eo1:enwiki-001298950027",
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