CS6840 - Modern Complexity Theory

Meetings  

• Theme 1 : Complexity Theory for Counting - 12 meetings

  • Meeting 01 : Thu, Jan 17, 11:00 am-11:50 am

  Administrative Announcements. Review of complexity classes, and the central questions of the area. Overview of the course. Counting Problems. Examples and algorithmic attempts. The class FP.

  References
  Exercises
  Reading

  Administrative Announcements. Review of complexity classes, and the central questions of the area. Overview of the course. Counting Problems. Examples and algorithmic attempts. The class FP.
  

  References

  :

  None

  • Meeting 02 : Fri, Jan 18, 10:00 am-10:50 am

  #CYCLE in FP then P=NP. The class #P. Examples. Containment of FP in #P.

  References	[AB] Section 9.1
  Exercises
  Reading

  #CYCLE in FP then P=NP. The class #P. Examples. Containment of FP in #P.
  

  References

  :

  [AB] Section 9.1

  • Meeting 03 : Mon, Jan 21, 08:00 am-08:50 am

  Bits of the #P function. The classes PP and Parity-P Characterization of #P vs FP question in the decision world using PP vs P question.

  References	[AB] Section 9.1
  Exercises
  Reading

  Bits of the #P function. The classes PP and Parity-P Characterization of #P vs FP question in the decision world using PP vs P question.
  

  References

  :

  [AB] Section 9.1

  • Meeting 04 : Tue, Jan 22, 12:00 pm-12:50 pm

  #P-hardness and completeness. Parsimonious reductions. #SAT is #P-complete.

  References	[AB] Section 9.1
  Exercises
  Reading

  #P-hardness and completeness. Parsimonious reductions. #SAT is #P-complete.
  

  References

  :

  [AB] Section 9.1

  • Meeting 05 : Thu, Jan 24, 11:00 am-11:50 am

  Determinant vs Permanent. Combinatorial Characterization of permanent of a matrix in terms of perfect matchings. Perm is in #P.

  References	[AB] Section 9.2 - refer to class notes for the exact sequence followed in the presentation. It is different from the textbook.
  Exercises
  Reading

  Determinant vs Permanent. Combinatorial Characterization of permanent of a matrix in terms of perfect matchings. Perm is in #P.
  

  References

  :

  [AB] Section 9.2 - refer to class notes for the exact sequence followed in the presentation. It is different from the textbook.

  • Meeting 06 : Fri, Jan 25, 10:00 am-10:50 am

  Characterization in terms of cycle covers. Reduction from #SAT to PERM over integers. Gadget construction. Cycle covers in the gadgets.

  References
  Exercises
  Reading

  Characterization in terms of cycle covers. Reduction from #SAT to PERM over integers. Gadget construction. Cycle covers in the gadgets.
  

  References

  :

  None

  • Meeting 07 : Mon, Jan 28, 08:00 am-08:50 am

  Counting the contribution to the cycle covers corresponding to satisfying assignments. Details of the construction.

  References
  Exercises
  Reading

  Counting the contribution to the cycle covers corresponding to satisfying assignments. Details of the construction.
  

  References

  :

  None

  • Meeting 08 : Tue, Jan 29, 12:00 pm-12:50 pm

  Reduction to -1,0,1 permanent. Reduction to 0-1 permanent. Using modulus arithmetic and using the polynomial method.

  References
  Exercises
  Reading

  Reduction to -1,0,1 permanent. Reduction to 0-1 permanent. Using modulus arithmetic and using the polynomial method.
  

  References

  :

  None

  • Meeting 09 : Thu, Jan 31, 11:00 am-11:50 am

  Question about the power of counting. Power of parity P. Attempt to include NP in parity P. Failures. The Valiant-Vazirani approach. The overview of the construction using hash families.

  References
  Exercises
  Reading

  Question about the power of counting. Power of parity P. Attempt to include NP in parity P. Failures. The Valiant-Vazirani approach. The overview of the construction using hash families.
  

  References

  :

  None

  • Meeting 10 : Fri, Feb 01, 10:00 am-10:50 am

  Valiant Vazirani Lemma. Details of the proof. Hash family construction. Amplification question.

  References
  Exercises
  Reading

  Valiant Vazirani Lemma. Details of the proof. Hash family construction. Amplification question.
  

  References

  :

  None

  • Meeting 11 : Mon, Feb 04, 08:00 am-08:50 am

  Amplifying the success probability for Parity SAT. The requirement of OR construction. Addition and Multiplication of Satisfying assignments.

  References
  Exercises
  Reading

  Amplifying the success probability for Parity SAT. The requirement of OR construction. Addition and Multiplication of Satisfying assignments.
  

  References

  :

  None

  • Meeting 12 : Tue, Feb 05, 12:00 pm-12:50 pm

  Generalization to PH. Eliminating randomization by counting. Toda's amplification lemma and modulus amplifying polynomials.

  References
  Exercises
  Reading

  Generalization to PH. Eliminating randomization by counting. Toda's amplification lemma and modulus amplifying polynomials.
  

  References

  :

  None

• Theme 2 : Complexity Theory for Randomization - 6 meetings

  • Meeting 13 : Thu, Feb 07, 11:00 am-11:50 am

  Randomized Algorithms and Counting Complexity. The class BPP. Examples.

  References	Class Notes.
  Exercises
  Reading

  Randomized Algorithms and Counting Complexity. The class BPP. Examples.
  

  References

  :

  Class Notes.

  • Meeting 14 : Fri, Feb 08, 10:00 am-10:50 am

  PIT problem and an example of a randomized algorithm. Schwartz-Zippel Lemma.

  References
  Exercises
  Reading

  PIT problem and an example of a randomized algorithm. Schwartz-Zippel Lemma.
  

  References

  :

  None

  • Meeting 15 : Tue, Feb 12, 12:00 pm-12:50 pm

  The class RP, relation with NP. Derandomization problem. Amplification of success probability.

  References
  Exercises
  Reading

  The class RP, relation with NP. Derandomization problem. Amplification of success probability.
  

  References

  :

  None

  • Meeting 16 : Thu, Feb 14, 11:00 am-11:50 am

  Two consequences of Amplification. BPP is in Sigma^2.

  References	Lecture Notes from the <a href="http://www.cse.iitm.ac.in/~jayalal/teaching/CS6840/2012/lecture11.pdf">past offering of this course</a>.
  Exercises
  Reading

  Two consequences of Amplification. BPP is in Sigma^2.
  

  References

  :

  Lecture Notes from the past offering of this course.

  • Meeting 17 : Fri, Feb 15, 10:00 am-10:50 am

  One random string for all. The class P/poly. BPP is in P/poly.

  References
  Exercises
  Reading

  One random string for all. The class P/poly. BPP is in P/poly.
  

  References

  :

  None

  • Meeting 18 : Fri, Feb 15, 02:00 pm-02:50 pm

  Karp-Lipton Collapse Theorem. If NP is contained P/poly then PH collapses down to Sigma_2

  References
  Exercises
  Reading

  Karp-Lipton Collapse Theorem. If NP is contained P/poly then PH collapses down to Sigma_2
  

  References

  :

  None

• Theme 3 : Circuit Complexity Theory - 27 meetings

  • Meeting 19 : Mon, Feb 18, 08:00 am-08:50 am

  Boolean circuits. And their connection to parallel algorithms. Example circuits, PARITY. Size and Depth. Trivial circuit for any function. The class PSIZE. Equivalence to P/poly.

  References
  Exercises
  Reading

  Boolean circuits. And their connection to parallel algorithms. Example circuits, PARITY. Size and Depth. Trivial circuit for any function. The class PSIZE. Equivalence to P/poly.
  

  References

  :

  None

  • Meeting 20 : Tue, Feb 19, 12:00 pm-12:50 pm

  The circuit lower bound problem. Shannon's Counting Argument. Discussion on explicitness.

  References
  Exercises
  Reading

  The circuit lower bound problem. Shannon's Counting Argument. Discussion on explicitness.
  

  References

  :

  None

  • Meeting 21 : Thu, Feb 21, 11:00 am-11:50 am

  Lupanov's circuit construction.

  References
  Exercises
  Reading

  Lupanov's circuit construction.
  

  References

  :

  None

  • Meeting 22 : Fri, Feb 22, 10:00 am-10:50 am

  Discussion on uniformity of circuit families. Characterizing P with P-Uniform PSIZE. Skew circuits. Characterizing NL with L-uniform SkewPSIZE. Variants and restrictions of cicuit complexity. Usefulness of a complexity theory in the circuits world.

  References
  Exercises
  Reading

  Discussion on uniformity of circuit families. Characterizing P with P-Uniform PSIZE. Skew circuits. Characterizing NL with L-uniform SkewPSIZE. Variants and restrictions of cicuit complexity. Usefulness of a complexity theory in the circuits world.
  

  References

  :

  None

  • Meeting 23 : Mon, Feb 25, 08:00 am-08:50 am

  Circuit design exercises. The PARITY example. Trivial constant depth unbounded fanin exp size circuit. Polynomial size linear depth bounded fanin circuit. Polynomial size log depth bounded fanin circuit. The constant depth question.<br><br> ADD(2,n) - adding two n-bit numbers. The ripple carry adder which gives linear size and linear depth circuits. The carry look ahead adder which gives constant depth unbounded fanin O(n^3) size circuits.<br><br> The class AC^0. <br><br> MULT(2,n). Iterated Addition ADD(n,n). Trivial log^2 depth circuit of bounded fanin and polysize. Improving depth to O(log n) using Offman's trick.

  References
  Exercises
  Reading

  Circuit design exercises. The PARITY example. Trivial constant depth unbounded fanin exp size circuit. Polynomial size linear depth bounded fanin circuit. Polynomial size log depth bounded fanin circuit. The constant depth question.
  
  ADD(2,n) - adding two n-bit numbers. The ripple carry adder which gives linear size and linear depth circuits. The carry look ahead adder which gives constant depth unbounded fanin O(n^3) size circuits.
  
  The class AC^0.
  
  MULT(2,n). Iterated Addition ADD(n,n). Trivial log^2 depth circuit of bounded fanin and polysize. Improving depth to O(log n) using Offman's trick.
  

  References

  :

  None

  • Meeting 24 : Tue, Feb 26, 12:00 pm-12:50 pm

  Iterated addition of log n, n-bit numbers is in AC^0.

  References
  Exercises
  Reading

  Iterated addition of log n, n-bit numbers is in AC^0.
  

  References

  :

  None

  • Meeting 25 : Thu, Feb 28, 11:00 am-11:50 am

  The class NC^1. ADD(n,n) is in NC^1. Classes AC^i and NC^i. Interleaving theorem. <br> The threshold functions. Majority. BITCOUNT. NC^1 upper bounds.

  References
  Exercises
  Reading

  The class NC^1. ADD(n,n) is in NC^1. Classes AC^i and NC^i. Interleaving theorem.
  The threshold functions. Majority. BITCOUNT. NC^1 upper bounds.
  

  References

  :

  None

  • Meeting 26 : Fri, Mar 01, 10:00 am-10:50 am

  MAJ constant depth reduces to MULT(2,n). BCount reduces to Threshold. Completing the class of reductions. The classes TC^0 and ACC^0.

  References
  Exercises
  Reading

  MAJ constant depth reduces to MULT(2,n). BCount reduces to Threshold. Completing the class of reductions. The classes TC^0 and ACC^0.
  

  References

  :

  None

  • Meeting 27 : Mon, Mar 04, 08:00 am-08:50 am

  Circuit Complexity Classes and Turing machine classes. <br> L-uniform NC^1 is contained in LOG.<br> NLOG is contained in L-uniform AC^1. The class SAC^1.

  References
  Exercises
  Reading

  Circuit Complexity Classes and Turing machine classes.
  L-uniform NC^1 is contained in LOG.
  NLOG is contained in L-uniform AC^1. The class SAC^1.
  

  References

  :

  None

  • Meeting 28 : Tue, Mar 05, 12:00 pm-12:50 pm

  Polynomial size Boolean Formulas and Equivalence to NC^1. Brent's construction.

  References
  Exercises
  Reading

  Polynomial size Boolean Formulas and Equivalence to NC^1. Brent's construction.
  

  References

  :

  None

  • Meeting 29 : Thu, Mar 07, 11:00 am-11:50 am

  Back to circuit lower bound problem. Gate Elimination argument. Th(n,2) requires 2n-4 gates over any basis. PARITY requires 3(n-1) AND-OR gates.

  References
  Exercises
  Reading

  Back to circuit lower bound problem. Gate Elimination argument. Th(n,2) requires 2n-4 gates over any basis. PARITY requires 3(n-1) AND-OR gates.
  

  References

  :

  None

  • Meeting 30 : Fri, Mar 08, 10:00 am-10:50 am

  Super linear lower bound for formulas - Sabbotavskaya. Superlinear lower bounds.

  References
  Exercises
  Reading

  Super linear lower bound for formulas - Sabbotavskaya. Superlinear lower bounds.
  

  References

  :

  None

  • Meeting 31 : Mon, Mar 11, 08:00 am-08:50 am

  Random restriction view of Sabbotavskaya's theorem. Shrinkage exponent.

  References
  Exercises
  Reading

  Random restriction view of Sabbotavskaya's theorem. Shrinkage exponent.
  

  References

  :

  None

  • Meeting 32 : Tue, Mar 12, 12:00 pm-12:50 pm

  Nechiporuk's Universal Functions. Super-linear lower bound.

  References
  Exercises
  Reading

  Nechiporuk's Universal Functions. Super-linear lower bound.
  

  References

  :

  None

  • Meeting 33 : Thu, Mar 14, 11:00 am-11:50 am

  Building on Nechiporuk's idea using Sabbotavskaya's theorem. Andreev's lower bound. Role of Shrinkage exponent.

  References
  Exercises
  Reading

  Building on Nechiporuk's idea using Sabbotavskaya's theorem. Andreev's lower bound. Role of Shrinkage exponent.
  

  References

  :

  None

  • Meeting 34 : Fri, Mar 15, 10:00 am-10:50 am

  Exponential Lower Bound for Constant Depth Circuits. Building a strategy for a lower bound. Measuring "simplifcation" under random restriction. Decision Trees. Canonical Decision trees from DNFs. Statement of the "DNF simplification theorem" (also known as switching lemma later).

  References
  Exercises
  Reading

  Exponential Lower Bound for Constant Depth Circuits. Building a strategy for a lower bound. Measuring "simplifcation" under random restriction. Decision Trees. Canonical Decision trees from DNFs. Statement of the "DNF simplification theorem" (also known as switching lemma later).
  

  References

  :

  None

  • Meeting 35 : Mon, Mar 18, 08:00 am-08:50 am

  Proof of main lower bound using the random restriction application lemma. Weaker bound first. Improving the weaker bound. Tightness of the lower bound by matching upper bound (from PS4).

  References
  Exercises
  Reading

  Proof of main lower bound using the random restriction application lemma. Weaker bound first. Improving the weaker bound. Tightness of the lower bound by matching upper bound (from PS4).
  

  References

  :

  None

  • Meeting 36 : Tue, Mar 19, 12:00 pm-12:50 pm

  Proof of the Switching Lemma. The definition of the bijection. The strategy of counting and recovery.

  References
  Exercises
  Reading

  Proof of the Switching Lemma. The definition of the bijection. The strategy of counting and recovery.
  

  References

  :

  None

  • Meeting 37 : Thu, Mar 21, 11:00 am-11:50 am

  References
  Exercises
  Reading

  
  

  References

  :

  None

  • Meeting 38 : Fri, Mar 22, 10:00 am-10:50 am

  Recovery procedure and the proof of the bijection. Combinatorial bounds for Stars(r,s). Completing the proof of the switching lemma.

  References
  Exercises
  Reading

  Recovery procedure and the proof of the bijection. Combinatorial bounds for Stars(r,s). Completing the proof of the switching lemma.
  

  References

  :

  None

  • Meeting 39 : Sat, Mar 23, 10:00 am-10:50 am

  Effect of the restriction on the circuit. The inductive lemma. Applying the switching lemma with appropriate parameters.

  References
  Exercises
  Reading

  Effect of the restriction on the circuit. The inductive lemma. Applying the switching lemma with appropriate parameters.
  

  References

  :

  None

  • Meeting 40 : Sat, Mar 23, 11:00 am-11:50 am

  Representing Boolean functions by polynomials. Multi-linearity property. Resources of degree and sparsity. Approximation by polynomials. Two methods. The notion of distance between a polynomial and a function. Outline that for constant depth circuit we expect a low degree polynomial approximating it. The case of OR function.

  References
  Exercises
  Reading

  Representing Boolean functions by polynomials. Multi-linearity property. Resources of degree and sparsity. Approximation by polynomials. Two methods. The notion of distance between a polynomial and a function. Outline that for constant depth circuit we expect a low degree polynomial approximating it. The case of OR function.
  

  References

  :

  None

  • Meeting 41 : Sat, Mar 30, 02:00 pm-02:50 pm

  Razborov Smolesnky method. Approximations by Polynomials. Building the approximator polynomial for a circuit. The induction step.

  References
  Exercises
  Reading

  Razborov Smolesnky method. Approximations by Polynomials. Building the approximator polynomial for a circuit. The induction step.
  

  References

  :

  None

  • Meeting 42 : Sat, Mar 30, 03:00 pm-03:50 pm

  Razborov Smolesnky method. Proof that Parity cannot be approximated by a small degree polynomial over F-3.

  References
  Exercises
  Reading

  Razborov Smolesnky method. Proof that Parity cannot be approximated by a small degree polynomial over F-3.
  

  References

  :

  None

  • Meeting 43 : Mon, Apr 01, 08:00 am-08:50 am

  Monotone Circuit Lower Bounds - I. Clique approximators, Approximation method, Positive and Negative inputs. Building the approximator from a circuit. The sunflower lemma.

  References
  Exercises
  Reading

  Monotone Circuit Lower Bounds - I. Clique approximators, Approximation method, Positive and Negative inputs. Building the approximator from a circuit. The sunflower lemma.
  

  References

  :

  None

  • Meeting 44 : Tue, Apr 02, 12:00 pm-12:50 pm

  Monotone Circuit Lower Bounds - II. Analysis of the errors made by the approximator. Size lower bound. Choice of parameters.

  References
  Exercises
  Reading

  Monotone Circuit Lower Bounds - II. Analysis of the errors made by the approximator. Size lower bound. Choice of parameters.
  

  References

  :

  None

  • Meeting 45 : Thu, Apr 04, 11:00 am-11:50 am

  Branching programs and the surprising theory by Barrington. NC^1 = Width-5 BPs.

  References
  Exercises
  Reading

  Branching programs and the surprising theory by Barrington. NC^1 = Width-5 BPs.
  

  References

  :

  None

• Theme 4 : Hardness vs Randomness - 6 meetings

  • Meeting 46 : Fri, Apr 05, 10:00 am-10:50 am

  To Be Announced

  References
  Exercises
  Reading

  To Be Announced
  

  References

  :

  None

  • Meeting 47 : Mon, Apr 08, 08:00 am-08:50 am

  To Be Announced

  References
  Exercises
  Reading

  To Be Announced
  

  References

  :

  None

  • Meeting 48 : Tue, Apr 09, 12:00 pm-12:50 pm

  To Be Announced

  References
  Exercises
  Reading

  To Be Announced
  

  References

  :

  None

  • Meeting 49 : Thu, Apr 11, 11:00 am-11:50 am

  To Be Announced

  References
  Exercises
  Reading

  To Be Announced
  

  References

  :

  None

  • Meeting 50 : Fri, Apr 12, 10:00 am-10:50 am

  To Be Announced

  References
  Exercises
  Reading

  To Be Announced
  

  References

  :

  None

  • Meeting 51 : Mon, Apr 15, 08:00 am-08:50 am

  To Be Announced

  References
  Exercises
  Reading

  To Be Announced
  

  References

  :

  None

• Theme 5 : Interactive Proofs & PCPs - 7 meetings

  • Meeting 52 : Tue, Apr 16, 12:00 pm-12:50 pm

  References
  Exercises
  Reading

  
  

  References

  :

  None

  • Meeting 53 : Thu, Apr 18, 11:00 am-11:50 am

  Viewing NP as an interactive proofs. Generalizations. Deterministic Interactive Proofs and characterizing NP using them. Role of randomness. The class IP. Interactive Proofs for GNI. Quest for characterizing the power of IP.

  References
  Exercises
  Reading

  Viewing NP as an interactive proofs. Generalizations. Deterministic Interactive Proofs and characterizing NP using them. Role of randomness. The class IP. Interactive Proofs for GNI. Quest for characterizing the power of IP.
  

  References

  :

  None

  • Meeting 54 : Fri, Apr 19, 10:00 am-10:50 am

  BPP is contained in IP. Role of randomness being private. Public Coin Protocols. AM and MA.<br> BPP is contained in MA.<br> Making MA protocols complete.

  References
  Exercises
  Reading

  BPP is contained in IP. Role of randomness being private. Public Coin Protocols. AM and MA.
  BPP is contained in MA.
  Making MA protocols complete.
  

  References

  :

  None

  • Meeting 55 : Mon, Apr 22, 08:00 am-08:50 am

  Making AM protocols complete. Replacing MA predicates with AM predicates. If all languages in CoNP had AM protocols then, so would all languages in Sigma_2.

  References
  Exercises
  Reading

  Making AM protocols complete. Replacing MA predicates with AM predicates. If all languages in CoNP had AM protocols then, so would all languages in Sigma_2.
  

  References

  :

  None

  • Meeting 56 : Tue, Apr 23, 12:00 pm-12:50 pm

  GNI is in AM. Set Lower Bound Protocol. Power of IP, Protocol for Permanent.

  References
  Exercises
  Reading

  GNI is in AM. Set Lower Bound Protocol. Power of IP, Protocol for Permanent.
  

  References

  :

  None

  • Meeting 57 : Thu, Apr 25, 11:00 am-11:50 am

  Protocol for PSPACE. IP=PSPACE.

  References
  Exercises
  Reading

  Protocol for PSPACE. IP=PSPACE.
  

  References

  :

  None

  • Meeting 58 : Fri, Apr 26, 10:00 am-10:50 am

  Approximation Algorithms, Inapproximability, PCP Theorem.

  References
  Exercises
  Reading

  Approximation Algorithms, Inapproximability, PCP Theorem.
  

  References

  :

  None