Ambrož Homar (2011) *The PCP theorem*. EngD thesis.

## Abstract

Solutions to NP-problems are deterministically verifiable in polynomial time. But the classic verification process has a drawback - typically we need to (at least) read the entire proof to decide whether the proof is correct or incorrect. In the thesis we present the PCP theorem which claims that solutions to NP- problems can be checked by only a small number of queries to bits in their corresponding proof strings. We describe the equivalent version of the theorem which is at the heart of many approximation results for NP-hard problems. We illustrate this on two well-known problems, the maximum independent set problem and the minimum vertex cover problem. We present the main ideas of both versions of the PCP theorem proof: the original algebraic proof and later combinatorial version. In the last chapter we list some exciting discoveries related to the probabilistic checking of proofs.

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