Abstract:
We describe a new proof of the PCP theorem that is based on a combinatorial amplification lemma. The *unsat value* of a set of constraints C={c_1,..,c_n}, denoted unsat(C), is the smallest fraction of unsatisfied constraints, ranging over all possible assignments for the underlying variables. We prove a new combinatorial amplification lemma that doubles the unsat-value of a constraint-system, with only a linear blowup in the size of the system. Iterative application of this lemma yields a proof for the PCP theorem. The amplification lemma relies on a new notion of "graph powering" that can be applied to systems of constraints. This powering amplifies the unsat-value of a constraint system provided that the underlying graph structure is an expander. We also apply the amplification lemma to construct PCPs and locally-testable codes whose length is linear up to a *polylog* factor, and whose correctness can be probabilistically verified by making a *constant* number of queries. Namely, we prove $SAT \in PCP_{\half,1}[\log_2(n\cdot\poly\log n),\,O(1)]$. This answers an open question of Ben-Sasson et al. (STOC '04).

Abstract:
Let C={c_1,...,c_n} be a set of constraints over a set of variables. The {\em satisfiability-gap} of C is the smallest fraction of unsatisfied constraints, ranging over all possible assignments for the variables. We prove a new combinatorial amplification lemma that doubles the satisfiability-gap of a constraint-system, with only a linear blowup in the size of the system. Iterative application of this lemma yields a self-contained (combinatorial) proof for the PCP theorem. The amplification lemma relies on a new notion of "graph powering" that can be applied to systems of constraints. This powering amplifies the satisfiability-gap of a constraint system provided that the underlying graph structure is an expander. We also apply the amplification lemma to construct PCPs and locally-testable codes whose length is {\em quasi-linear}, and whose correctness can be probabilistically verified by making a {\em constant} number of queries. Namely, we prove SAT \in PCP_{0.5,1}[log(n * polylog n) , O(1)]. This answers an open question of Ben-Sasson et al. (STOC '04).

Comment #1 to TR05-046 | 3rd June 2005 19:18

Gap Amplification Fails Below 1/2 Comment on: TR05-046

Abstract:
The gap amplification lemma of Dinur (ECCC TR05-46) states that the satisfiability gap of every d-regular constraint expander graph G (with self-loops) can be amplified by graph powering, as long as the satisfiability gap of G is not too large. We show that the last requirement is necessary. Namely, for infinitely many d and every t, there exists an integer n and a d-regular constraint expander G on n vertices over alphabet {0, 1} such that sat-gap(G) > 1/2 - o(d), but sat-gap(G^t) < 1/2.

Comment #2 to TR05-046 | 4th November 2005 09:38

Gap amplification using lazy random walks Comment on: TR05-046

Abstract:
This note is based on the original version of Irit Dinur's paper (ECCC TR05-046). It contains two suggestions concerning the product construction. First, instead of using paths of a fixed length t, one can use paths with varying lengths in order to simplify some calculations. Second, one can view Proposition 2.4 as guaranteeing some sort of pairwise independence, and use the second-moment method instead of explicitly bounding the overcount (as in Lemma 5.3).

Comment #3 to TR05-046 | 28th October 2007 18:32

A Small Gap in the Gap Amplification of Assignment Testers

Abstract:
An important extension of the proof of the PCP theorem by Irit Dinur (J. ACM 54(3), also ECCC TR05-046) is a gap amplification theorem for Assignment Testers. Specifically, this theorem states that the rejection probability of an Assignment Tester can be amplified by a constant factor, at the expense of increasing the output size of the Assignment Tester by a constant factor. We point out a gap in the proof of this theorem, and show that this gap can be filled.