Luca Trevisan.
Recycling Queries in PCPs and in Linearity Tests
We study query-efficient Probabilistically Checkable Proofs (PCPs) and linearity tests. We focus on the number of amortized query bits. A testing algorithm uses q amortized query bits if, for some constant k, it reads qk bits and has error probability at most 2-k. The best known PCP construction for NP in this respect uses 3 amortized query bits (Hastad, STOC97); at least one amortized query bit is necessary, unless P=NP (Bellare, Goldreich and Sudan, FOCS95). This parameter is an extremely natural one and has applications to proving non-approximability for constraint satisfaction problems. Furthermore, a PCP characterization of NP with less than 2 amortized query bits would imply a separation of the PCP model from the 2-Prover 1-Round model.

Our approach is to take an atomic verification procedure and then iterate it several times, saving queries by recycling them between different iterations of the atomic test.

We first apply this idea in order to develop query-efficient linearity tests. Linearity testing is a problem closely related to testing the Long Code and making PCP constructions. It is also a significant combinatorial problem still lacking tight characterizations, except for the case of three queries (Bellare et al. FOCS95). The best known linearity test uses 3 amortized query bits (Bellare et al., FOCS95); a different one achieves 1 amortized free bit (a different parameter related to the Max Clique problem) but uses an unbounded number of amortized query bits (Bellare, Goldreich and Sudan, FOCS95). We develop a general analysis technique and a linearity test achieving simultaneously amortized query complexity 1.5 and amortized free bit complexity 1/2. This test answers an open question raised by Bellare, Goldreich and Sudan.

We then show how to adapt a weaker result to the PCP setting, and we obtain a PCP for NP that makes 5 queries and has error probability 1/4, so that its amortized query complexity is 2.5.