Valiant–Vazirani theorem
E345812
The Valiant–Vazirani theorem is a fundamental result in computational complexity theory showing that solving unique solutions of NP problems is, under randomized reductions, as hard as solving general NP problems, with major implications for the study of randomness and hardness of approximation.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Valiant-Vazirani theorem | 1 |
| Valiant–Vazirani theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T3308898 — 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: Valiant–Vazirani theorem Context triple: [Leslie Valiant, knownFor, Valiant–Vazirani theorem]
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A.
Håstad’s switching lemma
Håstad’s switching lemma is a fundamental result in computational complexity theory that provides powerful bounds on the simplification of Boolean formulas under random restrictions, with major applications in circuit lower bounds.
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B.
Interactive Proofs and the Hardness of Approximating Cliques
"Interactive Proofs and the Hardness of Approximating Cliques" is a seminal theoretical computer science paper that introduced powerful interactive proof techniques to show that finding near-maximum cliques in graphs is computationally intractable to approximate within strong bounds.
-
C.
“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.
-
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.
The Knowledge Complexity of Interactive Proof Systems
"The Knowledge Complexity of Interactive Proof Systems" is a seminal theoretical computer science paper that introduced the notion of zero-knowledge proofs, fundamentally shaping modern cryptography and complexity theory.
- 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: Valiant–Vazirani theorem Target entity description: The Valiant–Vazirani theorem is a fundamental result in computational complexity theory showing that solving unique solutions of NP problems is, under randomized reductions, as hard as solving general NP problems, with major implications for the study of randomness and hardness of approximation.
-
A.
Håstad’s switching lemma
Håstad’s switching lemma is a fundamental result in computational complexity theory that provides powerful bounds on the simplification of Boolean formulas under random restrictions, with major applications in circuit lower bounds.
-
B.
Interactive Proofs and the Hardness of Approximating Cliques
"Interactive Proofs and the Hardness of Approximating Cliques" is a seminal theoretical computer science paper that introduced powerful interactive proof techniques to show that finding near-maximum cliques in graphs is computationally intractable to approximate within strong bounds.
-
C.
“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.
-
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.
The Knowledge Complexity of Interactive Proof Systems
"The Knowledge Complexity of Interactive Proof Systems" is a seminal theoretical computer science paper that introduced the notion of zero-knowledge proofs, fundamentally shaping modern cryptography and complexity theory.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
computational complexity theorem
ⓘ
result in theoretical computer science ⓘ |
| assumes | standard Turing machine model of computation ⓘ |
| canonicalReferenceTitle | NP is as easy as detecting unique solutions ⓘ |
| complexityAssumptionContext | P versus NP ⓘ |
| concernsProblem |
Unique-SAT
ⓘ
satisfiability problem ⓘ |
| field | computational complexity theory ⓘ |
| formalizes | randomized reduction from SAT to Unique-SAT ⓘ |
| hasImplicationFor |
derandomization
ⓘ
hardness of approximation ⓘ isolation lemma techniques ⓘ structure of NP ⓘ |
| implies | NP is contained in RP with an oracle for Unique-SAT ⓘ |
| influenced | isolation lemma in combinatorics and complexity ⓘ |
| isCentralTo |
study of promise problems in complexity theory
ⓘ
study of randomness in computation ⓘ |
| mainTopic |
NP complexity class
ⓘ
promise problems ⓘ randomized reductions ⓘ unique solutions of NP problems ⓘ |
| namedAfter |
Leslie Valiant
ⓘ
Vijay Vazirani ⓘ |
| originalAuthors |
Leslie Valiant
ⓘ
Vijay Vazirani ⓘ |
| originalPublicationType | journal article ⓘ |
| originalPublicationVenue | Theoretical Computer Science ⓘ |
| problemType | decision problem ⓘ |
| proofMethod | probabilistic method ⓘ |
| reductionType | randomized polynomial-time reduction ⓘ |
| relatesClass |
NP
ⓘ
PH ⓘ UP ⓘ ⊕P ⓘ |
| resultType | randomized reduction theorem ⓘ |
| shows |
If there is a polynomial-time algorithm for Unique-SAT then there is a randomized polynomial-time algorithm for SAT
ⓘ
Random hashing can isolate a unique satisfying assignment with non-negligible probability ⓘ Under randomized reductions, NP is no harder than detecting unique solutions in NP ⓘ |
| showsHardnessOf | Unique-SAT ⓘ |
| showsRelationshipBetween | search problems with many solutions and problems with a unique solution ⓘ |
| statesRoughly | Solving SAT instances with a unique satisfying assignment is as hard as solving general SAT under randomized polynomial-time reductions ⓘ |
| usedIn |
hardness results for unique games and related problems
ⓘ
reductions for approximate counting ⓘ |
| usesTechnique |
pairwise independent hash functions
ⓘ
random hashing ⓘ |
| yearProved | 1986 ⓘ |
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: Valiant–Vazirani theorem Description of subject: The Valiant–Vazirani theorem is a fundamental result in computational complexity theory showing that solving unique solutions of NP problems is, under randomized reductions, as hard as solving general NP problems, with major implications for the study of randomness and hardness of approximation.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject linked to:
Valiant
linked to: Valiant–Vazirani theorem