An Approach To The Sliding Scale Conjecture From Parallel Repetition for Low Degree Testing

Dana Moshkovitz, MIT

The Quest for Minimal Error

The Sliding Scale Conjecture Bellare, Goldwasser, Lund, Russell ’93

Every language in NP has a PCP verifier that uses r random bits and errs with probability =2-(r).

verifier

prover A prover B

Wish list:• Question size O(r).• Answer size O(log(1/)).• Randomness r=O(logn).

Implications to Hardness of Approximation

Hardness of n(1)-approximation for:• Max-CSP on polynomial sized alphabet.• Directed multi-cut, Directed sparsest cut

[Chuzhoy-Khanna].• Closest Vector Problem* [Arora, Babai, Stern,

Sweedyk]. (* assuming two provers, projection)

• Many more??

• 1992: constant error [Arora et al].Conjecture (Bellare et al): error = 2-(r).• 1994: Parallel repetition; two provers; randomness

(lognlog(1/)) [Raz].• 1997: = 2-(r1/3); five provers [Raz-Safra], [Arora-Sudan].• 1999: = 2- ( r1-); poly(1/) provers [Dinur et al]. • 2008: Two provers; randomness r=(1+o(1))logn; answer

size poly(1/) [M-Raz].• 2014: Parallel rep. of previous; two provers; randomness

r=O(logn); answer size poly(1/) [Dinur-Steurer].

History of Low Error

How To Get Low Error?

1. Algebraic, based on low degree testsStrong structural result, error not low enough

2. Combinatorial, based on parallel repetitionSize blow-up inherent [Feige-Kilian]

Our approach: parallel repetition for low degree testingstructure & lower error derandomization??

Line vs. Line Test

verifier

prover A prover B

x

Line through x

p2

univariate of deg d

p1 univariate of deg d

Line through x

p1(x)=p2(x)?

F – finite fieldm – dimensiond - degree

Thm (…,Arora-Sudan, 97): For sufficiently large field F wrt d, m, if P[line vs. line test passes], then there is a polynomial of degree at most d over Fm that agrees with - |F|-(1) of the lines.

Lemma 1: Low degree test with error |F|-(m), randomness O(mlog|F|), queries O(1), implies the Sliding Scale Conjecture.

Curve vs. Curve Test

verifier

prover A prover B

x1,…,xk’

Degree-k curve through x1,…,xk’

p2

univariate of deg dk

p1 univariate of deg dk

Degree-k curve through x1,…,xk’

i p1(xi)=p2(xi)?

More generally:

Surface vs. Surface

Problem: Provers Can Use Large Intersection To Cheat!

Per point x, provers decide on an m-variate polynomial Px of deg d.

prover A prover B

Restriction of Px’1

Restriction of Px1

x1, x2,…Points on A’s surface s1, sorted

x’1, x’2,…Points on B’s surface s2, sorted

With prob |s1s2|/|s1s2|: x1=x’1.

IKW Solution: Add Third Prover

verifier

prover A prover B

x1,…,xk’ x’1,…,x’k’

prover BA

Surface through x1,…,xk’

Surface through x1,…,xk’ x’1,…,x’k’

Surface through x’1,…,x’k’

Parallel Repetition for Low Degree Tests

Surface vs. Surface has error |F|-(1)

Repeated Test has error |F|-(k’).

Whereas IKW error 2-(k).

About Our Parallel Repetition Proof

• Not derandomized!• Improvement & simplification of IKW.• Gives structural guarantee (provers’ strategy agrees

with a polynomial) - used in proof.• Requires analysis of mixing properties of

incidence graphs.

Surface vs. Points Incidence Graph

• Bipartite graph on A={surfaces} and B={k’-tuples of points}; edges correspond to containment.

x1

x2

xk'

surfacesk‘-tuples

Identifying a Strategy

Surfaces through x

k‘-tuples through x

If the answers on surfaces are inconsistent on x…

as long as the graph is “mixing”, the inconsistency will get detected!

Mixing Parameters

Surfaces

Points in Fm

From k-wise independence: surfaces vs. points “mixes well”. What about surfaces vs. k’-tuples?

IKW: Extend k-wise independence argument, and get weak parameters.

Incidence Graphs As Product Graphs

surfacespoints in Fm

points in Fm

k‘ times

Surfaces through x

x

Product of mixing graphs also mixes.

Derandomized parallel repetition?

• Feige-Kilian: limitation on derandomizing parallel repetition.

• Avoided when the two k’-tuples in test are independent!

• Open: Can derandomize parallel repetition for low degree testing, and hence prove Sliding Scale Conjecture?

Challenge Problem: Intersecting Surfaces

Are there sized-|F|O(m+k’) families of surfaces and k’-tuples such that both the incidence graphs of surfaces vs. k’-tuples AND k’-tuples vs. points “mix well”, i.e.:

• Sampling: subset B’ of fraction of tuples, for fraction (1-) of surfaces s in A, fraction of the k’-tuples in s are in B’, for =|F|-(k’) and =|F|-(1).

• Dispersing: B’Fm, B’= |B|, for fraction at most of tuples s in A, we have sB’, for =|F|-(k’) and =|F|-(1).