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: We prove that all sets in $$\mathcal{NP}$$ 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 $$\mathcal{NP}$$ has a smooth strong canonical PCP of Proximity (PCPP), meaning that there is an efficiently computable bijection of $$\mathcal{NP}$$ 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.