An Approach To The Sliding Scale Conjecture Via Parallel Repetition For Low Degree Testing

The Sliding Scale Conjecture in PCP states that there are PCP verifiers with a constant number of queries and soundness error that is exponentially small in the randomness of the verifier and the length of the prover’s answers. The Sliding Scale Conjecture is one of the oldest open problems in PCP, and it implies hardness of approximation up to polynomial factors for problems like Max-CSP (with polynomial-sized alphabet), Directed-Sparsest-Cut and Directed-Multi-Cut. In this work we prove: 1. The Sliding Scale Conjecture follows from a construction of a low degree test whose soundness error is exponential in the randomness of the verifier. 2. A parallel repetition theorem for low degree testing: Given a low degree test with error |F|, one can generate a repeated low degree test whose error is |F|. 3. Applying the parallel repetition theorem on a suitable low degree test, we get a low degree test with error |F| and randomness O(km log |F|). In particular, we get the first low degree test with error ≪ 1/ |F| and O(m log |F|) randomness. The missing piece for proving the Sliding Scale Conjecture is a derandomization of the parallel repetition theorem. This seems plausible given the algebraic structure of the low degree testing problem, which was utilized for derandomization in the past. The limitation on derandomizing parallel repetition by Feige and Kilian does not rule out this approach. Additional contributions in this work include an analysis of the sampling properties of the incidence graph of degree-k curves and k′-tuples of points in a finite space F, and a combinatorial composition lemma for PCP that abstracts the composition technique of Arora and Safra. ∗[email protected]. Department of Electrical Engineering and Computer Science, MIT. This material is based upon work supported by the National Science Foundation under Grant Number 1218547.