Theory Seminar: Two Query PCP with Sub-Constant Error

Dana Moshkovitz, Weizmann

Sunday, 20.07.2008, 11:00

Taub 337

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 (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 (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 (Samorodnitsky and Trevisan).

4) Clique cannot be efficiently approximated to within a factor of n^{1-o(1)}, unless ZPP=NP. Previously, a hardness factor of n^{1-epsilon} for any constant epsilon>0 was known, under the assumption that P=NP does not hold (Hastad and Zuckerman).

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.

This is joint work with Ran Raz

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 (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 (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 (Samorodnitsky and Trevisan).

4) Clique cannot be efficiently approximated to within a factor of n^{1-o(1)}, unless ZPP=NP. Previously, a hardness factor of n^{1-epsilon} for any constant epsilon>0 was known, under the assumption that P=NP does not hold (Hastad and Zuckerman).

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.

This is joint work with Ran Raz