CS6850 - Topics in Complexity Theory

Meetings  

• Theme 1 : Algebraic Complexity Classes - 4 meetings

  • Meeting 01 : Sat, Jan 22, 10:00 am-12:00 pm - Jayalal

  Arithmetic Circuits. Algebraic Complexity Classes. Overview of structural results about circuits. Overview of lower bounds known for depth-restricted case, multilinear case, monotone case. The plan for the reading.

  References	[RP]
  Exercises
  Reading

  Arithmetic Circuits. Algebraic Complexity Classes. Overview of structural results about circuits. Overview of lower bounds known for depth-restricted case, multilinear case, monotone case. The plan for the reading.
  

  References

  :

  [RP]

  • Meeting 02 : Sat, Jan 29, 09:00 am-10:00 am - Jayalal

  Polynomials of Interest. Simpler models of circuits formulas Projections and completeness. Cycle cover based expression for permanent and determinants. Determinant and permanent are universal.

  References
  Exercises
  Reading

  Polynomials of Interest. Simpler models of circuits formulas Projections and completeness. Cycle cover based expression for permanent and determinants. Determinant and permanent are universal.
  

  References

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  None

  • Meeting 03 : Sat, Jan 29, 10:00 am-11:00 am - Jayalal

  VNP-completeness Completeness of the permanent. Rosette and the connecting gadgets and cycle covers.

  References
  Exercises
  Reading

  VNP-completeness Completeness of the permanent. Rosette and the connecting gadgets and cycle covers.
  

  References

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  None

  • Meeting 04 : Sat, Feb 05, 09:00 am-09:55 am - Jayalal

  Details of the VNP-completeness. Algebraic Branching Programs. ABPs for IMM. ABP for determinant using claw sequence algorithm. From ABPs to determinants.

  References
  Exercises
  Reading

  Details of the VNP-completeness. Algebraic Branching Programs. ABPs for IMM. ABP for determinant using claw sequence algorithm. From ABPs to determinants.
  

  References

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  None

• Theme 2 : Structural Results - 2 meetings

  • Meeting 05 : Sat, Feb 05, 10:00 am-11:00 am - Neha

  Homogenization. Computing homogeneous components. Interpolation technique. Depth reduction for arithmetic formulas. Brent's Construction.

  References
  Exercises
  Reading

  Homogenization. Computing homogeneous components. Interpolation technique. Depth reduction for arithmetic formulas. Brent's Construction.
  

  References

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  None

  • Meeting 06 : Sat, Feb 05, 11:00 am-12:30 pm - Neha

  Depth reduction for arithmetic circuits. Brent's construction based on degree. VSBR construction. Reduction to depth four.

  References
  Exercises
  Reading

  Depth reduction for arithmetic circuits. Brent's construction based on degree. VSBR construction. Reduction to depth four.
  

  References

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  None

• Theme 3 : Classical Lower Bounds - 4 meetings

  • Meeting 07 : Sat, Feb 12, 10:00 am-11:00 am - Bhabya

  Baur-Strassen Bound. Computing partial derivatives.

  References
  Exercises
  Reading

  Baur-Strassen Bound. Computing partial derivatives.
  

  References

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  None

  • Meeting 08 : Sat, Feb 12, 11:00 am-12:00 pm - Bhabya

  Kalarkoti's quadratic lower bound for formulas.

  References
  Exercises
  Reading

  Kalarkoti's quadratic lower bound for formulas.
  

  References

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  None

  • Meeting 09 : Sat, Feb 19, 10:00 am-12:00 pm - Jayalal

  Determinantal Complexity Lower Bounds. Mignon-Rossayre bound.

  References
  Exercises
  Reading

  Determinantal Complexity Lower Bounds. Mignon-Rossayre bound.
  

  References

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  None

  • Meeting 10 : Sat, Feb 26, 10:00 am-11:00 am - Neha/Bhabya

  Natural Proof Strategies. Monotone Circuits.

  References
  Exercises
  Reading

  Natural Proof Strategies. Monotone Circuits.
  

  References

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  None

• Theme 4 : Depth Three Circuits - 3 meetings

  • Meeting 11 : Sat, Feb 26, 11:00 am-12:00 pm - Neha/Bhabya

  Partial derivatives. Low-rank depth-3 circuits.

  References
  Exercises
  Reading

  Partial derivatives. Low-rank depth-3 circuits.
  

  References

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  None

  • Meeting 12 : Sat, Mar 05, 10:00 am-12:00 pm -

  Nisan-Wigderson Lower bounds for depth three. Sym-d has large space of partial derivatives.

  References
  Exercises
  Reading

  Nisan-Wigderson Lower bounds for depth three. Sym-d has large space of partial derivatives.
  

  References

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  None

  • Meeting 13 : Sat, Mar 05, 10:00 am-12:00 pm -

  Shpilka-Wigderson Lower bound. Grigoriev-Karpinski Lower bound (to be planned).

  References
  Exercises
  Reading

  Shpilka-Wigderson Lower bound. Grigoriev-Karpinski Lower bound (to be planned).
  

  References

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  None

• Theme 5 : Multilinear Circuits - 2 meetings

  • Meeting 14 : Sat, Mar 12, 10:00 am-12:00 pm -

  The Partial Derivative Matrix method. Det and Perm have large rank. Constructing a full-rank polynomial.

  References
  Exercises
  Reading

  The Partial Derivative Matrix method. Det and Perm have large rank. Constructing a full-rank polynomial.
  

  References

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  None

  • Meeting 15 : Sat, Mar 19, 10:00 am-12:00 pm -

  Separating multilinear ABPs and formulas

  References
  Exercises
  Reading

  Separating multilinear ABPs and formulas
  

  References

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  None

• Theme 6 : Monotone Circuits - 1 meetings

  • Meeting 16 : Sat, Mar 26, 10:00 am-12:00 pm -

  Separating monotone circuits and monotone ABPs. The separation between monotone VP and VNP (if time permits).

  References
  Exercises
  Reading

  Separating monotone circuits and monotone ABPs. The separation between monotone VP and VNP (if time permits).
  

  References

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  None

• Theme 7 : Depth Four Circuits - 2 meetings

  • Meeting 17 : Sat, Apr 02, 10:00 am-12:00 pm -

  Lower bounds for depth-4 circuits with bounded bottom fan-in

  References
  Exercises
  Reading

  Lower bounds for depth-4 circuits with bounded bottom fan-in
  

  References

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  None

  • Meeting 18 : Sat, Apr 09, 10:00 am-12:00 pm -

  Lower bounds for homogeneous depth four circuits

  References
  Exercises
  Reading

  Lower bounds for homogeneous depth four circuits
  

  References

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  None

• Theme 8 : Constant Depth Circuits - 1 meetings

  • Meeting 19 : Sat, Apr 16, 10:00 am-12:00 pm -

  Lower bounds for constant depth circuits

  References
  Exercises
  Reading

  Lower bounds for constant depth circuits
  

  References

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  None

• Theme 9 : Limitations of current techniques - 1 meetings

  • Meeting 20 : Sat, Apr 23, 10:00 am-12:00 pm -

  Limitations of sub-additive rank methods

  References
  Exercises
  Reading

  Limitations of sub-additive rank methods
  

  References

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  None