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It runs in polynomial time on inputs that are in SUBSET-SUM if and only if P = NP: // Algorithm that accepts the NP-complete language SUBSET-SUM. // // this is a polynomial-time algorithm if and only if P = NP. // // "Polynomial-time" means it returns "yes" in polynomial time when // the answer should be "yes", and runs forever when it is "no".
A propositional proof system is given as a proof-verification algorithm P(A,x) with two inputs.If P accepts the pair (A,x) we say that x is a P-proof of A.P is required to run in polynomial time, and moreover, it must hold that A has a P-proof if and only if A is a tautology.
For instance, we say that query evaluation has polynomial-time data complexity for a class of queries if, for every fixed query Q in that class, given a database D, we can compute the answers to Q on D in polynomial time. Less commonly, we can study the query complexity, which is the computational complexity when the database is fixed and when ...
If there is a polynomial-time algorithm for even one of them, then there is a polynomial-time algorithm for all the problems in NP. Because of this, and because dedicated research has failed to find a polynomial algorithm for any NP-complete problem, once a problem has been proven to be NP-complete, this is widely regarded as a sign that a ...
Propositional proof system can be compared using the notion of p-simulation. A propositional proof system P p-simulates Q (written as P ≤ p Q) when there is a polynomial-time function F such that P(F(x)) = Q(x) for every x. [1] That is, given a Q-proof x, we can find in polynomial time a P-proof of the same tautology.
A problem is polynomial-time Turing-reducible to a problem if, given a subroutine that solves in polynomial time, one could write a program that calls this subroutine and solves in polynomial time. This contrasts with many-one reducibility, which has the restriction that the program can only call the subroutine once, and the return value of the ...
The PCP theorem states that NP = PCP[O(log n), O(1)],. where PCP[r(n), q(n)] is the class of problems for which a probabilistically checkable proof of a solution can be given, such that the proof can be checked in polynomial time using r(n) bits of randomness and by reading q(n) bits of the proof, correct proofs are always accepted, and incorrect proofs are rejected with probability at least 1/2.
The algorithm can check in polynomial time if the vertices in G appear once in c. Additionally, it takes polynomial time to check the start and end vertices, as well as the edges between vertices. Therefore, the algorithm is a polynomial time verifier for the Hamiltonian path problem. [22]