We show that the NP-Complete language 3Sat has a PCP
verifier that makes two queries to a proof of almost-linear size
and achieves sub-constant probability of error $o(1)$. The
verifier performs only projection tests, meaning that the answer
to the first query determines at most one accepting answer to the
second query.
Previously, by the parallel repetition theorem, there were PCP
Theorems with two-query projection tests, but only (arbitrarily
small) constant error, and polynomial size.
There were also PCP Theorems with sub-constant error and
almost-linear size, but a constant number of queries that is
larger than $2$.
As a corollary, we obtain a host of new results. In particular,
our theorem improves many of the hardness of approximation
results that are proved using the parallel repetition theorem. A
partial list includes the following:
1) 3Sat cannot be efficiently approximated to within
a factor of 7/8+o(1), unless P=NP. This holds
even under almost-linear reductions. Previously, the best known
NP-hardness factor was 7/8+ epsilon for any
constant epsilon>0, under polynomial reductions
(by Hastad).
2) 3Lin cannot be efficiently approximated to within
a factor of 1/2+o(1), unless P=NP. This holds
even under almost-linear reductions. Previously, the best known
NP-hardness factor was 1/2+epsilon for any
constant epsilon>0, under polynomial reductions
(by Hastad).
3) A PCP Theorem with amortized query complexity 1 + o(1) and
amortized free bit complexity o(1). Previously, the best known
amortized query complexity and free bit complexity were
1+epsilon and epsilon, respectively, for any constant
epsilon > 0 (by Samorodnitsky and Trevisan).
One of the new ideas that we use is a new technique for doing the
composition step in the (classical) proof of the PCP
Theorem, without increasing the number of queries to the proof. We
formalize this as a composition of new objects that we call
Locally Decode/Reject Codes (LDRC). The notion of LDRC was
implicit in several previous works, and we make it explicit in
this work. We believe that the formulation of LDRCs and their
construction are of independent interest.
