Karp reductions
E519560
concept in computational complexity theory
polynomial-time many-one reduction
reduction between decision problems
Karp reductions are polynomial-time many-one reductions used in computational complexity theory to show that one decision problem is at least as hard as another, central to defining NP-completeness.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Karp reductions canonical | 1 |
| Levin reduction | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5429797 — 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: Karp reductions Context triple: [Richard Karp, notableConcept, Karp reductions]
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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.
Cook–Levin theorem
The Cook–Levin theorem is a foundational result in computational complexity theory that established the Boolean satisfiability problem (SAT) as the first NP-complete problem, launching the theory of NP-completeness.
-
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.
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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.
-
E.
Blum complexity measures
Blum complexity measures are a formal framework in computational complexity theory that rigorously define and compare the resource usage (such as time or space) of algorithms via axiomatic conditions.
- 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: Karp reductions Target entity description: Karp reductions are polynomial-time many-one reductions used in computational complexity theory to show that one decision problem is at least as hard as another, central to defining NP-completeness.
-
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.
Cook–Levin theorem
The Cook–Levin theorem is a foundational result in computational complexity theory that established the Boolean satisfiability problem (SAT) as the first NP-complete problem, launching the theory of NP-completeness.
-
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.
“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.
Blum complexity measures
Blum complexity measures are a formal framework in computational complexity theory that rigorously define and compare the resource usage (such as time or space) of algorithms via axiomatic conditions.
- F. None of above. chosen
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
concept in computational complexity theory
ⓘ
polynomial-time many-one reduction ⓘ reduction between decision problems ⓘ |
| alsoKnownAs |
P-time many-one reduction
ⓘ
polynomial-time many-one reduction ⓘ |
| assumptionOnEncoding | instances are encoded as finite strings over a fixed alphabet ⓘ |
| canonicalExampleFrom | SAT ⓘ |
| canonicalExampleTo |
CLIQUE
ⓘ
HAMILTONIAN CYCLE ⓘ
linked to:
Hamiltonian cycle concept
VERTEX COVER ⓘ |
| category | many-one reduction ⓘ |
| closureProperty | transitive ⓘ |
| codomain | decision problem ⓘ |
| complexityClassContext |
NP
ⓘ
NP-complete ⓘ P ⓘ |
| computabilityRequirement | reduction function must be computable in polynomial time ⓘ |
| contrastWith |
Cook reduction
ⓘ
Turing reduction ⓘ
linked to:
Turing reducibility
|
| domain | decision problem ⓘ |
| ensuresProperty | if B is in P and A ≤_m^P B then A is in P ⓘ |
| field | computational complexity theory ⓘ |
| formalDefinition | a function f from instances of problem A to instances of problem B such that x is in A if and only if f(x) is in B and f is computable in polynomial time ⓘ |
| impliesHardness | if A Karp-reduces to B and A is NP-hard then B is NP-hard ⓘ |
| importance | standard notion of reduction for NP-completeness proofs ⓘ |
| influencedBy | earlier notions of reducibility in recursion theory ⓘ |
| introducedBy |
Richard M. Karp
ⓘ
linked to:
Richard Karp
|
| introducedInWork | "Reducibility Among Combinatorial Problems" ⓘ |
| introducedInYear | 1972 ⓘ |
| logicalForm | single call transformation from one instance to another ⓘ |
| mappingType | many-one reduction ⓘ |
| namedAfter |
Richard M. Karp
ⓘ
linked to:
Richard Karp
|
| NPCompleteDefinitionRole | a problem is NP-complete if it is in NP and every problem in NP Karp-reduces to it ⓘ |
| preservesMembership | x ∈ A iff f(x) ∈ B ⓘ |
| problemType | decision problems with yes/no answers ⓘ |
| relatedConcept |
Cook–Levin theorem
ⓘ
complete problems for NP ⓘ polynomial-time reduction ⓘ |
| relationToNPCompleteness | used to define NP-complete problems ⓘ |
| requires | polynomially bounded output size ⓘ |
| strongerThan | polynomial-time Turing reduction in terms of restriction ⓘ |
| symbolicNotation | ≤_m^P ⓘ |
| timeComplexityConstraint | polynomial time ⓘ |
| transitivityDescription | if A ≤_m^P B and B ≤_m^P C then A ≤_m^P C ⓘ |
| usedFor |
comparing hardness of decision problems
ⓘ
proving NP-completeness ⓘ showing one problem is at least as hard as another ⓘ |
| usedIn |
classification of NP-complete problems
ⓘ
reductions between SAT and other problems ⓘ |
| usedToShow | NP-hardness of optimization problems via decision versions ⓘ |
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: Karp reductions Description of subject: Karp reductions are polynomial-time many-one reductions used in computational complexity theory to show that one decision problem is at least as hard as another, central to defining NP-completeness.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Karp reductions