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

The Quest for Minimal Error. An Approach To The Sliding Scale Conjecture From Parallel Repetition for Low Degree Testing. Dana Moshkovitz , MIT. The Sliding Scale Conjecture Bellare , Goldwasser , Lund, Russell ’93.

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

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  1. The Quest for Minimal Error An Approach To The Sliding Scale Conjecture From Parallel Repetition for Low Degree Testing Dana Moshkovitz, MIT

  2. 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). prover A prover B • Wish list: • Question size O(r). • Answer size O(log(1/)). • Randomness r=O(logn). verifier

  3. 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??

  4. History of Low Error • 1992: constant error [Arora et al]. Conjecture (Bellare et al): error  =2-(r). • 1994: Parallel repetition; two provers; randomness (lognlog(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].

  5. How To Get Low Error? • Algebraic, based on low degree tests Strong structural result, error not low enough • Combinatorial, based on parallel repetition Size blow-up inherent [Feige-Kilian] Our approach: parallel repetition for low degree testing structure & lower error derandomization??

  6. F – finite field m – dimension d - degree Line vs. Line Test p1univariate of deg d p2 univariate of deg d p1(x)=p2(x)? x prover A prover B Line through x Line through x verifier

  7. Thm (…,Arora-Sudan, 97): For sufficiently large field Fwrtd, 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.

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

  9. More generally: Surface vs. Surface Curve vs. Curve Test p1univariate of deg dk p2 univariate of deg dk ip1(xi)=p2(xi)? x1,…,xk’ prover A prover B Degree-k curve through x1,…,xk’ Degree-k curve through x1,…,xk’ verifier

  10. Problem: Provers Can Use Large Intersection To Cheat! Per point x, provers decide on an m-variate polynomial Px of deg d. Restriction of Px1 Restriction of Px’1 prover A prover B With prob|s1s2|/|s1s2|: x1=x’1. x1, x2,… x’1, x’2,… Points on A’s surface s1, sorted Points on B’s surface s2, sorted

  11. IKW Solution: Add Third Prover Surface through x1,…,xk’ x’1,…,x’k’ Surface through x’1,…,x’k’ Surface through x1,…,xk’ x1,…,xk’ x’1,…,x’k’ prover A prover B prover BA verifier

  12. Parallel Repetition for Low Degree Tests Surface vs. Surface has error |F|-(1)  Repeated Test has error |F|-(k’). Whereas IKW  error 2-(k).

  13. 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.

  14. Surface vs. Points Incidence Graph xk' • Bipartite graph on A={surfaces} and B={k’-tuples of points}; edges correspond to containment. x2 x1 surfaces k‘-tuples 

  15. 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!

  16. Mixing Parameters From k-wise independence: surfaces vs. points “mixes well”. What about surfaces vs. k’-tuples? Surfaces Points in Fm IKW: Extend k-wise independence argument, and get weak parameters. 

  17. Incidence Graphs As Product Graphs Product of mixing graphs also mixes. surfaces points in Fm  Surfaces through x x  points in Fm k‘ times

  18. 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?

  19. 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 tupless in A, we have sB’, for =|F|-(k’) and =|F|-(1).

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