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748:{\displaystyle \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 )}}
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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
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254:{\displaystyle \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})}
1079:{\displaystyle \max \limits _{S_{1},\dots ,S_{\ell }}|\{(v_{1},\dots ,v_{m})\colon \phi (R_{1},\dots ,R_{k},S_{1},\dots ,S_{\ell },v_{1},\dots ,v_{m})\}|}
377:
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
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and at most 3 literals per clause), find an assignment satisfying as many clauses as possible. In fact, it is a natural
72:
1430:
1293:
Papadimitriou, Christos H.; Yannakakis, Mihalis (1991). "Optimization, approximation, and complexity classes".
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is defined as the class of optimization problems on relational structures expressible in the following form:
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correspond to sets of vertices colored with one of the 3 colors. Similarly, SNP contains the
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1314:
1236:
Feder, Tomás; Vardi, Moshe Y. (1993). "Monotone monadic SNP and constraint satisfaction".
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For example, SNP contains 3-Coloring (the problem of determining whether a given graph is
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tuples, one wants to maximize the number of tuples for which it is satisfied. That is,
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Proceedings of the twenty-fifth annual ACM symposium on Theory of computing - STOC '93
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denotes the adjacency relation of the input graph, while the sets (unary relations)
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are relations of the structure (such as the adjacency relation, for a graph),
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1195:-complete (under PTAS reductions), and hence does not admit a PTAS unless
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17:
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1347:. Texts in Theoretical Computer Science. An EATCS Series. Berlin:
56:
properties. It forms the basis for the definition of the class
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It is defined as the class of problems that are properties of
388:), because it can be expressed by the following formula:
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are unknown relations (sets of tuples of vertices), and
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1167:(slightly more general than L-reductions) is equal to
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is then defined as the class of all problems with an
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1343:Finite model theory and its applications
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1122:: given an instance of 3-CNF-SAT (the
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46:containing a limited subset of
32:computational complexity theory
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1179:to it from some problem in
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847:literals per clause, where
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1145:to solve any problem in
1183:. In particular, every
1143:approximation algorithm
1141:There is a fixed-ratio
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369:{\displaystyle \phi }
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69:relational structures
62:optimization problems
1295:J. Comput. Syst. Sci
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1126:with the formula in
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1102:log-space reduction
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1408:Complexity Zoo
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1397:Complexity Zoo
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1386:Complexity Zoo
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1379:External links
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1329:Libkin, Leonid
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1301:(3): 425–440.
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1201:PCP theorem
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386:3-colorable
1371:1133.03001
1351:. p.
1315:0765.68036
1257:0897915917
1223:References
1191:) is also
851:is fixed.
1274:cite book
1050:…
1029:ℓ
1018:…
986:…
967:ϕ
964::
945:…
914:ℓ
903:…
714:¬
711:∨
689:¬
682:∧
654:¬
651:∨
629:¬
620:∧
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591:∨
569:¬
561:⟹
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439:∀
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412:∃
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342:ℓ
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212:ℓ
201:…
169:…
150:ϕ
136:∀
133:…
120:∀
114:ℓ
106:∃
103:…
90:∃
71:(such as
40:Strict NP
1425:Category
1211:See also
1149:, hence
1132:complete
1113:MAX-3SAT
1266:9229294
42:) is a
18:MAX-SNP
1413:MaxSNP
1402:MaxSNP
1369:
1359:
1313:
1264:
1254:
1205:MaxSNP
1185:MaxSNP
1181:MaxSNP
1175:has a
1163:under
1161:MaxSNP
1151:MaxSNP
1147:MaxSNP
1136:MaxSNP
1117:MaxSNP
1106:MaxSNP
1100:, not
1090:MaxSNP
869:MaxSNP
855:MaxSNP
264:where
73:graphs
58:MaxSNP
38:(from
1262:S2CID
758:Here
1357:ISBN
1280:link
1252:ISBN
1197:P=NP
1391:SNP
1367:Zbl
1353:350
1311:Zbl
1303:doi
1242:doi
1217:APX
1193:APX
1173:APX
1169:APX
1156:APX
886:max
865:all
60:of
36:SNP
30:In
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1365:.
1355:.
1335:;
1309:.
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