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An additive combinatorics approach to the log-rank conjecture in communication complexity

Noga Zewi Technion – Israel Institute of Technology. An additive combinatorics approach to the log-rank conjecture in communication complexity. Joint work with: Eli Ben- Sasson ( Technion and MSR New England) Shachar Lovett ( IAS, Princeton). Communication Complexity. …. Protocol P.

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An additive combinatorics approach to the log-rank conjecture in communication complexity

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  1. NogaZewi Technion – Israel Institute of Technology An additive combinatorics approach to the log-rank conjecture in communication complexity Joint work with: Eli Ben-Sasson (Technion and MSR New England) Shachar Lovett (IAS, Princeton)

  2. Communication Complexity … Protocol P CC(P) = Max total # of bits sent on worst case inputs. CC(f) =

  3. Log-rank Conjecture Fact. Log-rank conjecture. [Lovász, Saks, FOCS 88]

  4. Known Bounds Upper bounds: Lower bounds: [Kotlov, Lovász 96] [Nisan, Wigderson 95] [Kotlov 97] [Kushilevitz 95] Main result. Assuming the polynomial Freiman-Ruzsa conjecture,

  5. Polynomial Freiman-Ruzsa Conjecture • |A+A|=|A|  A is a subspace  |span(A)|=|A| Question: |A+A|/|A| is small => |span(A)|/|A| is small? Thm. [Ruzsa 99] Polynomial Freiman-Ruzsa (PFR) Conjecture. integer r: Thm.[Even-Zohar 11] Thm. [Sanders 10]

  6. Proof Overview Main tool: Approximte duality [Ben-Sasson, Z. 11, construction of two-source extractors] PFR Conjecture Second part of talk Main technical contribution Improved bounds on approximate duality Uses methodology Suggested by Nisan,Wigderson 95 First part of talk Communication complexity upper bounds

  7. Approximate Duality Duality measure: ? Question: => large Thm. [Ben-Sasson, Z. 11] Main technical lemma. (Assuming PFR) (Assuming PFR)

  8. From Approximate Duality to CC Upper Bounds Discrepancy of a 0-1 matrix M: • => M is monochromatic. • => M is balanced. Thm 1. [NW95]Nisan-Wigderson Conj.  Log-rank Conj. Nisan-Wigderson Conjecture. [NW95] submatrix of , Thm 2. [NW95, spectral methods] submatrix of , We show: Approximate duality => submatrix of , => matrix , .

  9. From Approximate Duality to CC upper bounds Observation 1. Enough to assume and work over ! Observation 2. Goal: submatrixof , => (approximate duality) => Conclusion: is a submatrix of ,

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