Smooth and Strong PCPs

Probabilistically checkable proofs (PCPs) can be verified based only on a constant amount of random queries, such that any correct claim has a proof that is always accepted, and incorrect claims are rejected with high probability (regardless of the given alleged proof). We consider two possible features of PCPs: $$\circ \quad$$ ∘ A PCP is strong if it rejects an alleged proof of a correct claim with probability proportional to its distance from some correct proof of that claim. $$\circ \quad$$ ∘ A PCP is smooth if each location in a proof is queried with equal probability. We prove that all sets in $$\mathcal{NP}$$ NP have PCPs that are both smooth and strong, are of polynomial length and can be verified based on a constant number of queries. This is achieved by following the proof of the PCP theorem of Arora et al. (JACM 45(3):501–555, 1998), providing a stronger analysis of the Hadamard and Reed–Muller based PCPs and a refined PCP composition theorem. In fact, we show that any set in $$\mathcal{NP}$$ NP has a smooth strong canonical PCP of Proximity (PCPP), meaning that there is an efficiently computable bijection of $$\mathcal{NP}$$ NP witnesses to correct proofs. This improves on the recent construction of Dinur et al. (in: Blum (ed) 10th innovations in theoretical computer science conference, ITCS, San Diego, 2019) of PCPPs that are strong canonical but inherently non-smooth. Our result implies the hardness of approximating the satisfiability of “ stable ” 3CNF formulae with bounded variable occurrence , where stable means that the number of clauses violated by an assignment is proportional to its distance from a satisfying assignment (in the relative Hamming metric). This proves a hypothesis used in the work of Friggstad, Khodamoradi and Salavatipour (in: Chan (ed) Proceedings of the 30th annual ACM-SIAM symposium on discrete algorithms, SODA, San Diego, 2019), suggesting a connection between the hardness of these instances and other stable optimization problems.