NP-hardness
E512971
NP-hardness is a classification in computational complexity theory for problems at least as hard as the hardest problems in NP, such that every problem in NP can be reduced to them in polynomial time.
All labels observed (1)
| Label | Occurrences |
|---|---|
| NP-hardness canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5335895 — 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-hardness Context triple: [P, NP, and NP-Completeness: The Basics of Complexity Theory, mainTopic, NP-hardness]
-
A.
NP-completeness
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.
-
B.
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.
-
C.
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.
-
D.
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.
-
E.
“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.
- 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-hardness Target entity description: NP-hardness is a classification in computational complexity theory for problems at least as hard as the hardest problems in NP, such that every problem in NP can be reduced to them in polynomial time.
-
A.
NP-completeness
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.
-
B.
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.
-
C.
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.
-
D.
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.
-
E.
“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.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf | computational complexity theory concept ⓘ |
| appliesTo | decision versions of optimization problems ⓘ |
| assumes | polynomial-time computability of reductions ⓘ |
| characterization | at least as hard as every problem in NP ⓘ |
| closureProperty | closed under polynomial-time reductions ⓘ |
| complexityStatus | captures worst-case difficulty relative to NP ⓘ |
| consequence |
if any NP-hard problem is in P then P = NP
ⓘ
polynomial-time algorithm for one NP-complete problem yields polynomial-time algorithms for all NP problems ⓘ |
| definedOver |
decision problems
ⓘ
optimization problems ⓘ search problems ⓘ |
| distinguishedFrom |
NP-completeness
ⓘ
NP-easiness ⓘ co-NP-hardness ⓘ |
| doesNotRequire | membership in NP ⓘ |
| exampleProblem |
3-SAT is NP-hard
ⓘ
linked to:
3-SAT
clique problem (optimization version) is NP-hard ⓘ halting problem (under suitable reductions) ⓘ
linked to:
Halting problem
satisfiability problem for Boolean formulas (SAT) is NP-hard ⓘ subset sum optimization problem is NP-hard ⓘ traveling salesman problem (optimization version) is NP-hard ⓘ
linked to:
traveling salesman problem
vertex cover problem (optimization version) is NP-hard ⓘ |
| field |
computational complexity theory
ⓘ
linked to:
Complexity Theory
theoretical computer science ⓘ |
| formalDefinition | a problem H is NP-hard if for every problem L in NP, L ≤p H ⓘ |
| generalizationOf | NP-completeness ⓘ |
| historicalWork |
Cook–Levin theorem
ⓘ
Karp's 21 NP-complete problems paper (1972) ⓘ |
| implies | problem is at least as hard as NP-complete problems ⓘ |
| introducedBy |
Richard Karp
ⓘ
Stephen Cook ⓘ |
| keyCondition | every problem in NP reduces to the problem in polynomial time ⓘ |
| reductionType |
polynomial-time Turing reduction
ⓘ
polynomial-time many-one reduction ⓘ |
| relatedClass |
EXPTIME
ⓘ
NP ⓘ NP-completeness ⓘ PSPACE ⓘ co-NP ⓘ |
| subsetRelation |
NP-complete problems are NP-hard and in NP
ⓘ
NP-hard problems may lie outside NP ⓘ |
| typicalAssumption | P ≠ NP implies no polynomial-time algorithm for NP-hard problems ⓘ |
| typicalUseContext | reductions in hardness proofs ⓘ |
| usedFor |
classifying computational intractability
ⓘ
proving non-existence of efficient algorithms unless P = NP ⓘ |
| variant |
strong NP-hardness
ⓘ
weak NP-hardness ⓘ |
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-hardness Description of subject: NP-hardness is a classification in computational complexity theory for problems at least as hard as the hardest problems in NP, such that every problem in NP can be reduced to them in polynomial time.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.