In computational complexity theory, the PCP theorem (also known as the PCP characterization theorem) states that every decision problem in the NP complexity class has probabilistically checkable proofs (proofs that can be checked by a randomized algorithm) of constant query complexity and logarithmic randomness complexity (uses a logarithmic number of random bits). The PCP theorem says that for some universal constant K {\displaystyle K} , for every n {\displaystyle n} , any mathematical proof for a statement of length n {\displaystyle n} can be rewritten as a different proof of length poly ( n ) {\displaystyle \operatorname {poly} (n)} that is formally verifiable with 99% accuracy by a randomized algorithm that inspects only K {\displaystyle K} letters of that proof. The PCP theorem is the cornerstone of the theory of computational hardness of approximation, which investigates the inherent difficulty in designing efficient approximation algorithms for various optimization problems. It has been described by Ingo Wegener as "the most important result in complexity theory since Cook's theorem" and by Oded Goldreich as "a culmination of a sequence of impressive works […] rich in innovative ideas".
Formal statement The PCP theorem states that N P = P C P [ O ( log n ) , O ( 1 ) ] {\displaystyle {\mathsf {NP}}={\mathsf {PCP}}[O(\log n),O(1)]}
where N P {\displaystyle {\mathsf {NP}}} is the complexity class of problems solvable in nondeterministic polynomial time and where P C P [ r ( n ) , q ( n ) ] {\displaystyle {\mathsf {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 ) {\displaystyle r(n)} bits of randomness and by reading q ( n ) {\displaystyle q(n)} bits of the proof, correct proofs are always accepted, and incorrect proofs are rejected with probability at least 1 2 {\displaystyle {\tfrac {1}{2}}} . The variable n {\displaystyle n} is the length in bits of the description of a problem instance. Note further that the verification algorithm is non-adaptive: the choice of bits of the proof to check depend only on the random bits and the description of the problem instance, not the actual bits of the proof.
PCP and hardness of approximation An alternative formulation of the PCP theorem states that the maximum fraction of satisfiable constraints of a certain constraint satisfaction problem is NP-hard to approximate within some constant factor. Formally, for some constants q {\displaystyle q} and α < 1 {\displaystyle \alpha <1} , the following promise problem ( L y e s , L n o ) {\displaystyle (L_{\mathrm {yes} },L_{\mathrm {no} })} is an NP-hard decision problem:
L y e s = { Φ : {\displaystyle L_{\mathrm {yes} }=\{\Phi :} all constraints in Φ {\displaystyle \Phi } are simultaneously satisfiable } {\displaystyle \}}
L n o = { Φ : {\displaystyle L_{\mathrm {no} }=\{\Phi :} every assignment satisfies fewer than an α {\displaystyle \alpha } fraction of Φ {\displaystyle \Phi } 's constraints } {\displaystyle \}}
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