PCP theorem

The PCP theorem states that \mathsf{NP}=\mathsf{PCP}[O(\log n),O(1)] where \mathsf{NP} is the complexity class of problems solvable in nondeterministic polynomial time and where \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) 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 \tfrac12. The variable 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 and \alpha, the following promise problem (L_{\mathrm{yes}},L_{\mathrm{no}}) is an NP-hard decision problem: • L_{\mathrm{yes}}=\{\Phi: all constraints in \Phi are simultaneously satisfiable\} • L_{\mathrm{no}}=\{\Phi: every assignment satisfies fewer than an \alpha fraction of \Phi's constraints\} where \Phi is a constraint satisfaction problem (CSP) over a Boolean alphabet with at most q variables per constraint. The connection to the class \mathsf{PCP} mentioned above can be seen by noticing that checking a constant number of bits q in a proof can be seen as evaluating a constraint in q Boolean variables on those bits of the proof. Since the verification algorithm uses O(\log n) bits of randomness, it can be represented as a CSP as described above with \operatorname{poly}(n) constraints. The first formulation of the PCP theorem then guarantees the promise condition with \alpha=\tfrac12: if the NP problem's answer is yes, then every constraint (which corresponds to a particular value for the random bits) has a satisfying assignment (an acceptable proof); otherwise, any proof should be rejected with probability at least \tfrac12, which means any assignment must satisfy fewer than \tfrac12 of the constraints (which means it will be accepted with probability lower than \tfrac12). Therefore, an algorithm for the promise problem would be able to solve the underlying NP problem, and hence the promise problem must be NP-hard. As a consequence of this theorem, it can be shown that the solutions to many natural optimization problems including maximum boolean formula satisfiability, maximum independent set in graphs, and the shortest vector problem for lattices cannot be approximated efficiently unless \mathsf{P}=\mathsf{NP}. This can be done by reducing the problem of approximating a solution to such problems to a promise problem of the above form. These results are sometimes also called PCP theorems because they can be viewed as probabilistically checkable proofs for NP with some additional structure. == Proof ==

A proof of a weaker result, {{tmath|\mathsf{NP} \subseteq \mathsf{PCP}[n^3, 1]}} is given in one of the lectures of Dexter Kozen. ==History==

The PCP theorem is the culmination of a long line of work on interactive proofs and probabilistically checkable proofs. The first theorem relating standard proofs and probabilistically checkable proofs is the statement that \mathsf{NEXP}\subseteq\mathsf{PCP}[\operatorname{poly}(n),\operatorname{poly}(n)], proved by . Origin of the initials The notation \mathsf{PCP}_{c(n),s(n)}[r(n),q(n)] is explained at probabilistically checkable proof. The notation is that of a function that returns a certain complexity class. See the explanation mentioned above. The name of this theorem (the "PCP theorem") probably comes either from "PCP" meaning "probabilistically checkable proof", or from the notation mentioned above (or both). First theorem [in 1990] Subsequently, the methods used in this work were extended by Babai, Lance Fortnow, Levin, and Szegedy in 1991 , Feige, Goldwasser, Lund, Safra, and Szegedy (1991), and Arora and Safra in 1992 to yield a proof of the PCP theorem by Arora, Lund, Motwani, Sudan, and Szegedy in 1998 . The 2001 Gödel Prize was awarded to Sanjeev Arora, Uriel Feige, Shafi Goldwasser, Carsten Lund, László Lovász, Rajeev Motwani, Shmuel Safra, Madhu Sudan, and Mario Szegedy for work on the PCP theorem and its connection to hardness of approximation. In 2005 Irit Dinur discovered a significantly simpler proof of the PCP theorem, using expander graphs. She received the 2019 Gödel Prize for this. == Quantum analogs ==

Nonlocal games version A version of the PCP theorem for quantum nonlocal games would state that it is computationally hard to approximate the quantum value of a quantum nonlocal game. In 2012, Thomas Vidick and Tsuyoshi Ito published a result that showed a "strong limitation on the ability of entangled provers to collude in a multiplayer game". This could be a step toward proving the quantum analogue of the PCP theorem, since when the result In 2018, Thomas Vidick and Anand Natarajan proved a games variant of quantum PCP theorem under randomized reduction. It states that \mathsf{QMA}\subseteq\mathsf{MIP}^*[\log n,1,\tfrac12], where \mathsf{MIP}^*[f(n),c,s] is a complexity class of multi-prover quantum interactive proofs systems with f(n)-bit classical communications, and the completeness is c and the soundness is s. Hamiltonian version A version of the PCP theorem for quantum local Hamiltonians would state that it is computationally hard to approximate the ground energy of a quantum local Hamiltonian. The work of Natarajan and Vidick The conjecture was proven in 2022 by Anurag Anshu, Nikolas Breuckmann, and Chinmay Nirkhe using a construction of a Hamiltonian based on quantum CSS codes. However, the ground states of such Hamiltonians are given explicitly by the code states of the corresponding CSS code, and thus are not computationally hard enough for a general proof of a quantum PCP theorem. ==Notes==