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Mahaney's theorem
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Mahaney's theorem
Mahaney's theorem is a theorem in computational complexity theory proven by Stephen Mahaney that states that:
Note that the existence of an NP-hard sparse set implies the existence of an NP-complete sparse set.
Mahaney's theorem was motivated by the Berman–Hartmanis conjecture:
Since we know that some NP-complete sets are not sparse (for example, the set of satisfiable 3SAT formulas), Berman and Hartmanis derived a second, weaker conjecture:
Mahaney's result shows that, if P ≠ NP, then indeed there are no NP-complete sparse sets, thus settling the second conjecture under the standard assumption of P ≠ NP.
The result was strengthened in 1991 to state that:
This is stronger than Mahaney's theorem, which is the special case where the polynomial-time algorithm can make at most 1 query to the sparse language protocol.
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Mahaney's theorem
Mahaney's theorem is a theorem in computational complexity theory proven by Stephen Mahaney that states that:
Note that the existence of an NP-hard sparse set implies the existence of an NP-complete sparse set.
Mahaney's theorem was motivated by the Berman–Hartmanis conjecture:
Since we know that some NP-complete sets are not sparse (for example, the set of satisfiable 3SAT formulas), Berman and Hartmanis derived a second, weaker conjecture:
Mahaney's result shows that, if P ≠ NP, then indeed there are no NP-complete sparse sets, thus settling the second conjecture under the standard assumption of P ≠ NP.
The result was strengthened in 1991 to state that:
This is stronger than Mahaney's theorem, which is the special case where the polynomial-time algorithm can make at most 1 query to the sparse language protocol.