NP-completeness
E173625
NP-completeness is a central concept in computational complexity theory that classifies decision problems believed to be among the hardest in NP, such that a polynomial-time solution to any one of them would yield polynomial-time solutions to all problems in NP.
All labels observed (4)
| Label | Occurrences |
|---|---|
| NP-completeness canonical | 4 |
| Cook–Levin theorem | 1 |
| Karp’s 21 NP-complete problems | 1 |
| theory of NP-completeness | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1531384 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: NP-completeness Context triple: [Introduction to Algorithms, coversTopic, NP-completeness]
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A.
P, NP, and NP-Completeness: The Basics of Complexity Theory
"P, NP, and NP-Completeness: The Basics of Complexity Theory" is a foundational textbook by Oded Goldreich that introduces the core concepts, problems, and techniques of computational complexity theory, with a focus on the classes P, NP, and NP-complete problems.
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B.
PCP theorem
The PCP theorem is a fundamental result in computational complexity theory stating that every problem in NP has probabilistically checkable proofs that can be verified by examining only a constant number of bits, with major implications for the hardness of approximation.
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C.
P versus NP problem
The P versus NP problem is a central unsolved question in theoretical computer science that asks whether every problem whose solution can be quickly verified by a computer can also be quickly solved by a computer.
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D.
“Inapproximability results for SAT and other problems”
“Inapproximability results for SAT and other problems” is a seminal theoretical computer science paper by Johan Håstad that establishes tight hardness-of-approximation bounds for satisfiability and related optimization problems using probabilistically checkable proofs.
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E.
Computational Complexity: A Conceptual Perspective
Computational Complexity: A Conceptual Perspective is a graduate-level textbook that presents the foundations and key themes of computational complexity theory with an emphasis on conceptual understanding over technical detail.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: NP-completeness Target entity description: NP-completeness is a central concept in computational complexity theory that classifies decision problems believed to be among the hardest in NP, such that a polynomial-time solution to any one of them would yield polynomial-time solutions to all problems in NP.
-
A.
P, NP, and NP-Completeness: The Basics of Complexity Theory
"P, NP, and NP-Completeness: The Basics of Complexity Theory" is a foundational textbook by Oded Goldreich that introduces the core concepts, problems, and techniques of computational complexity theory, with a focus on the classes P, NP, and NP-complete problems.
-
B.
PCP theorem
The PCP theorem is a fundamental result in computational complexity theory stating that every problem in NP has probabilistically checkable proofs that can be verified by examining only a constant number of bits, with major implications for the hardness of approximation.
-
C.
P versus NP problem
The P versus NP problem is a central unsolved question in theoretical computer science that asks whether every problem whose solution can be quickly verified by a computer can also be quickly solved by a computer.
-
D.
“Inapproximability results for SAT and other problems”
“Inapproximability results for SAT and other problems” is a seminal theoretical computer science paper by Johan Håstad that establishes tight hardness-of-approximation bounds for satisfiability and related optimization problems using probabilistically checkable proofs.
-
E.
Computational Complexity: A Conceptual Perspective
Computational Complexity: A Conceptual Perspective is a graduate-level textbook that presents the foundations and key themes of computational complexity theory with an emphasis on conceptual understanding over technical detail.
- F. None of above. chosen
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
complexity class property
ⓘ
computational complexity theory concept ⓘ |
| assumesConjecture | P ≠ NP (widely believed) ⓘ |
| centralReference |
Garey and Johnson: Computers and Intractability
ⓘ
Papadimitriou: Computational Complexity ⓘ Introduction to the Theory of Computation ⓘ
surface form:
Sipser: Introduction to the Theory of Computation
|
| characterizes | hardest problems in NP ⓘ |
| closureProperty | closed under polynomial-time many-one reductions ⓘ |
| concerns | worst-case time complexity ⓘ |
| consequenceOfIntractability | if all NP-complete problems are intractable then P ≠ NP ⓘ |
| consequenceOfPTimeSolution | P = NP if any NP-complete problem has a polynomial-time algorithm ⓘ |
| definedOver | decision problems ⓘ |
| dependsOn |
definition of NP
ⓘ
nondeterministic Turing machine model ⓘ polynomial-time computability ⓘ |
| field |
computational complexity theory
ⓘ
theoretical computer science ⓘ |
| firstNPCompleteProblem |
Cook–Levin theorem
ⓘ
surface form:
Boolean satisfiability problem
|
| firstNPCompleteProof | Cook–Levin theorem ⓘ |
| formalDefinition | A decision problem L is NP-complete if L is in NP and every problem in NP is polynomial-time reducible to L ⓘ |
| hasCanonicalProblem |
3-SAT
ⓘ
Clique problem ⓘ Hamiltonian cycle problem ⓘ SAT ⓘ Subset sum problem ⓘ Traveling salesman decision problem ⓘ Vertex cover problem ⓘ |
| hasVariant |
#P-completeness
ⓘ
PSPACE-completeness ⓘ co-NP-completeness ⓘ parameterized NP-completeness (W[1]-hardness etc.) ⓘ |
| implies |
NP-hardness
ⓘ
membership in NP ⓘ problem is at least as hard as every problem in NP ⓘ |
| introducedBy |
Leonid Levin
ⓘ
Stephen Cook ⓘ |
| introductionYear | 1971 ⓘ |
| oftenAnalyzedWith | reduction chains between problems ⓘ |
| relatedTo |
NP
ⓘ
NP-hardness ⓘ P versus NP problem ⓘ |
| requires |
NP-hardness
ⓘ
belonging to NP ⓘ |
| studiedIn | complexity theory textbooks ⓘ |
| typicalAssumption | NP-complete problems do not admit polynomial-time algorithms ⓘ |
| typicalProofMethod | polynomial-time reduction from known NP-complete problem ⓘ |
| usedFor |
classifying computational intractability
ⓘ
showing hardness of approximation in some contexts ⓘ |
| usesReductionType |
Karp reduction
ⓘ
polynomial-time many-one reduction ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: NP-completeness Description of subject: NP-completeness is a central concept in computational complexity theory that classifies decision problems believed to be among the hardest in NP, such that a polynomial-time solution to any one of them would yield polynomial-time solutions to all problems in NP.
Referenced by (7)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Cook–Levin theorem
this entity surface form:
theory of NP-completeness
this entity surface form:
Karp’s 21 NP-complete problems
subject surface form:
Introduction to Automata Theory, Languages, and Computation