Meetings

Click on the theme item for the meeting plan for that theme.Click on the meeting item for references, exercises, and additional reading related to it.

**Theme 1 : Complexity Theory for Counting**- 12 meetings- Meeting 01 : Thu, Jan 17, 11:00 am-11:50 am
- Meeting 02 : Fri, Jan 18, 10:00 am-10:50 am
- Meeting 03 : Mon, Jan 21, 08:00 am-08:50 am
- Meeting 04 : Tue, Jan 22, 12:00 pm-12:50 pm
- Meeting 05 : Thu, Jan 24, 11:00 am-11:50 am
- Meeting 06 : Fri, Jan 25, 10:00 am-10:50 am
- Meeting 07 : Mon, Jan 28, 08:00 am-08:50 am
- Meeting 08 : Tue, Jan 29, 12:00 pm-12:50 pm
- Meeting 09 : Thu, Jan 31, 11:00 am-11:50 am
- Meeting 10 : Fri, Feb 01, 10:00 am-10:50 am
- Meeting 11 : Mon, Feb 04, 08:00 am-08:50 am
- Meeting 12 : Tue, Feb 05, 12:00 pm-12:50 pm

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 #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 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 #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 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. 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 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 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 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 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 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 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
- Meeting 14 : Fri, Feb 08, 10:00 am-10:50 am
- Meeting 15 : Tue, Feb 12, 12:00 pm-12:50 pm
- Meeting 16 : Thu, Feb 14, 11:00 am-11:50 am
- Meeting 17 : Fri, Feb 15, 10:00 am-10:50 am
- Meeting 18 : Fri, Feb 15, 02:00 pm-02:50 pm

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. 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 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 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. 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 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_2References : None **Theme 3 : Circuit Complexity Theory**- 27 meetings- Meeting 19 : Mon, Feb 18, 08:00 am-08:50 am
- Meeting 20 : Tue, Feb 19, 12:00 pm-12:50 pm
- Meeting 21 : Thu, Feb 21, 11:00 am-11:50 am
- Meeting 22 : Fri, Feb 22, 10:00 am-10:50 am
- Meeting 23 : Mon, Feb 25, 08:00 am-08:50 am
- Meeting 24 : Tue, Feb 26, 12:00 pm-12:50 pm
- Meeting 25 : Thu, Feb 28, 11:00 am-11:50 am
- Meeting 26 : Fri, Mar 01, 10:00 am-10:50 am
- Meeting 27 : Mon, Mar 04, 08:00 am-08:50 am
- Meeting 28 : Tue, Mar 05, 12:00 pm-12:50 pm
- Meeting 29 : Thu, Mar 07, 11:00 am-11:50 am
- Meeting 30 : Fri, Mar 08, 10:00 am-10:50 am
- Meeting 31 : Mon, Mar 11, 08:00 am-08:50 am
- Meeting 32 : Tue, Mar 12, 12:00 pm-12:50 pm
- Meeting 33 : Thu, Mar 14, 11:00 am-11:50 am
- Meeting 34 : Fri, Mar 15, 10:00 am-10:50 am
- Meeting 35 : Mon, Mar 18, 08:00 am-08:50 am
- Meeting 36 : Tue, Mar 19, 12:00 pm-12:50 pm
- Meeting 37 : Thu, Mar 21, 11:00 am-11:50 am
- Meeting 38 : Fri, Mar 22, 10:00 am-10:50 am
- Meeting 39 : Sat, Mar 23, 10:00 am-10:50 am
- Meeting 40 : Sat, Mar 23, 11:00 am-11:50 am
- Meeting 41 : Sat, Mar 30, 02:00 pm-02:50 pm
- Meeting 42 : Sat, Mar 30, 03:00 pm-03:50 pm
- Meeting 43 : Mon, Apr 01, 08:00 am-08:50 am
- Meeting 44 : Tue, Apr 02, 12:00 pm-12:50 pm
- Meeting 45 : Thu, Apr 04, 11:00 am-11:50 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 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 Lupanov's circuit construction. References Exercises Reading Lupanov's circuit construction.References : None 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 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 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 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 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 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 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 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 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 Random restriction view of Sabbotavskaya's theorem. Shrinkage exponent. References Exercises Reading Random restriction view of Sabbotavskaya's theorem. Shrinkage exponent.References : None Nechiporuk's Universal Functions. Super-linear lower bound. References Exercises Reading Nechiporuk's Universal Functions. Super-linear lower bound.References : None 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 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 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 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 References Exercises Reading References : None 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 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 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 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 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 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 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 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
- Meeting 47 : Mon, Apr 08, 08:00 am-08:50 am
- Meeting 48 : Tue, Apr 09, 12:00 pm-12:50 pm
- Meeting 49 : Thu, Apr 11, 11:00 am-11:50 am
- Meeting 50 : Fri, Apr 12, 10:00 am-10:50 am
- Meeting 51 : Mon, Apr 15, 08:00 am-08:50 am

To Be Announced References Exercises Reading To Be AnnouncedReferences : None To Be Announced References Exercises Reading To Be AnnouncedReferences : None To Be Announced References Exercises Reading To Be AnnouncedReferences : None To Be Announced References Exercises Reading To Be AnnouncedReferences : None To Be Announced References Exercises Reading To Be AnnouncedReferences : None To Be Announced References Exercises Reading To Be AnnouncedReferences : None **Theme 5 : Interactive Proofs & PCPs**- 7 meetings- Meeting 52 : Tue, Apr 16, 12:00 pm-12:50 pm
- Meeting 53 : Thu, Apr 18, 11:00 am-11:50 am
- Meeting 54 : Fri, Apr 19, 10:00 am-10:50 am
- Meeting 55 : Mon, Apr 22, 08:00 am-08:50 am
- Meeting 56 : Tue, Apr 23, 12:00 pm-12:50 pm
- Meeting 57 : Thu, Apr 25, 11:00 am-11:50 am
- Meeting 58 : Fri, Apr 26, 10:00 am-10:50 am

References Exercises Reading References : None 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 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 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 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 Protocol for PSPACE. IP=PSPACE. References Exercises Reading Protocol for PSPACE. IP=PSPACE.References : None Approximation Algorithms, Inapproximability, PCP Theorem. References Exercises Reading Approximation Algorithms, Inapproximability, PCP Theorem.References : None