Computational Complexity Theory

Success probablity amplification of randomized algorithms using expander graphs. Definitions of imperfect randomness. Basic set up for derandomization. Concept of Extractors.

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Suggested Additional Reading:

Spectral gap for any regular graph. Equivalence between expansion definitions. Expander mixing lemma. Graph products and details of Reingold's log-space algorithm for testing undirected connectivity.

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Suggested Additional Reading:

Random walks on graphs. Mixing lemma. Randomized log space algorithm for undirected connectivity. Spectral expansion, combinatorial expansion. Basic definitions and parameters. Log space connectivity for expander graphs. Outline of Reingold's algorithm.

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Suggested Additional Reading:

BPP in \Sigma_2 \cap \Pi_2. Space bounded randomized classes.

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Suggested Additional Reading:

Randomized Computation and Complexity Classes PP, BPP and RP. Comparison of these classes with the ones we have seen so far like P, NP, PSPACE etc. Amplifying the success probability. Complete derandomization. BPP \in P/poly. RP algorithm for Polynomial identity testing problem.

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Suggested Additional Reading:

Power of Negations. Markov's tight characterisation for number of negation gates that are needed to compute any Boolean function. Fisher's strengthening to polynomial size setting. A method to derive lower bounds against a negation limited circuits from lower bounds against monotone circuits.

References:

• Mystery of Negations in Computing by Stasys Jukna.
• Chapter 6 of the new (April 27, 2009 !) upcoming book
  Frontiers in Boolean Function Complexity by Stasys Jukna.
• Polynomial size monotone circuit/formulae for threshold function :
  Lecture 11-12 in Uri Zwick's course on Concrete Complexity :

Suggested Additional Reading:

• A Note on the Inversion Complexity of Boolean Functions in Boolean Formulas - Hiroki Morizumi.
• A Superpolynomial Lower Bound for a Circuit Computing the Clique Function with At Most (1/6) log log n Negation Gates - Kazuyuki Amano and Akira Maruoka (1998)
• On the minimum number of negations leading to super-polynomial savings -- Stasys Jukna (2004)

Monotone Boolean functions and monotone Boolean circuits. Clique function. Upper bounds on monotone circuit size computing clique. Almost tight (which is super polynomial) lower bounds for monotone circuit size for clique.

References:

• Lecture Notes from V. Arvind's course, written by Ramprasad Saptarishi.
• Lecture 7 and Lecture 8 from Uri Zwick's course.

Suggested Additional Reading:

• Monotone Circuits for Matching Require Linear Depth - Raz and Widgerson.
• Monotone Circuits for Connectivity Require Super-logarithmic Depth - Karchmer and wigderson.

Representation and Approximation of Boolean functions by polynomials. Smolensky's polynomial method. Proof that PARITY is not in AC^0[3] (AC^0 circuits which can also use mod-3 gates).

References:

• Lecture 6 from Madhu Sudan's course.
• Lecture 17 from Peter Bro Miltersen's course.
• Lecture 10 from Kristoffer Hansen's Course.

Suggested Additional Reading:

• The Polynomial Method in Circuit Complexity - Richard Beigel
• Correlation bounds for polynomials over {0,1} - Emanuele Viola.

Review of last lecture. Rest of the details of the proof of lower bounds for parity. Proof of Switching Lemma.

References:

• Lecture 15 from Peter Bro Miltersen's course.
• (page 1 to 7 and page 18-20) of A Switching Lemma Primer - Paul Beame.
• Lecture 9 from Kristoffer Hansen's Course.

Suggested Additional Reading:

• Lecture 5 from Madhu Sudan's course has a slightly different (possibly simpler to understand) proof of what we did in class.

Explicit Lower Bound question. Review of decision tree model, depth/height lower bounds for parity. Exponential size lower bounds for constant depth circuits computing parity. Statement of switching Lemma. Completing the proof assuming switching lemma. Proof of switching lemma postponed to next lecture.

References:

• Lecture 15 from Peter Bro Miltersen's course.
• (page 1 to 7 and page 18-20) of A Switching Lemma Primer - Paul Beame.
• Lecture 9 from Kristoffer Hansen's Course.

Suggested Additional Reading:

• Lecture 5 from Madhu Sudan's course has a slightly different (possibly simpler to understand) proof of what we did in class.

Advice Functions, P/poly, Sparse sets. Turing reduction to Sparse sets. Circuit Lower bounds. Conditional Circuit Lower bounds for NP (Karp-Lipton-Sipser Collapse result).
Shannon's theorem (almost all Boolean functions require 2^n/n size circuits).
Lypanov's theorem (all Boolean function are computed by circuits of size 2^n/n+o(2^n/n)), proof skipped.

References:

• Chapter 1, of Vollmer's book on "Introduction to Circuit Complexity".
• Chapter 4 of Jin Yi Cai's full set of Lecture Notes.

Suggested Additional Reading:

• Reviewing bounds on the circuit size of the hardest functions.
  - Gudmund S. Frandsen and Peter Bro Miltersen.

Poly size Boolean Formulas computes exactly NC^1. Decision tree codel of computation, upper and lower bounds for parity function. Deterministic, Non-deterministic and permutation branching programs. Characterisations for L, NL. Programs over algebraic structures. Barrington's Theorem.

References:

• Lecture 12 in Jin Yi Cai's course.
• BWBP = NC^1 - David Barrington.

Suggested Additional Reading:

• Finite Monoids and Fine Structure of NC^1 - David Barrington and Denis Therien
• Circuits on Cylinders - Kristoffer Arnsfelt Hansen, Peter Bro Miltersen and V Vinay.

Back to Structural Complexity : Formal treatment of Polynomial Heirarchy, Oracle Turing Machines. Alternating Turing Machines, Basic Simulations. Circuit characterisation.

References:

• Lecture in Jin Yi Cai's course.
• Chapter 3 of the book on "Theory of Computational Complexity" - Du & Ko

Suggested Additional Reading:

TC^0 circuits for symmetric functions. Adding n numbers of length n each (ITADD - Iterated Addition) in NC^1. Adding log n numbers of length n each (LOGITADD - Log Iterated Addition) in AC^0. NC^1 circuit for Majority. AC^0 circuits for log Threshold computation.

References:

• Chapter 1, of Vollmer's book on "Introduction to Circuit Complexity".

Suggested Additional Reading:

Majority function. Definitions of AC, NC, ACC, TC heirarchy.
Computing threshold in NC^1 (outline) and logarithmic threshold in AC^0 (outline).

References:

Suggested Additional Reading:

Nonuniform models of computation. Boolean Circuit Complexity. Resources in the circuit model. Computing partity function in O(log n) depth and bounded fan-in. Definition of NC^1. Addition of two n-bit numbers in constant depth and unbounded fan-in. Definition of AC^0, ACC^0. Uniformity constraints. Comparison with Turing machine model.

References:

Suggested Additional Reading:

• Attend ITCS Seminar by Kristoffer Hansen on Thrusday at 4pm. The topic is "Constant Width Characterizations of Small Depth Circuit Classes" and will talk about characterisations of NC^1, AC^0, ACC^0 that we mentioned in the class.

Immerman-Szelepcsenyi Theorem. A short description of oracle turing machines and notion of relativisation, and the metalevel argument about relativising proof techniques. Polynomial Hierarchy (postponed to next lecture).

References: Chapter 2 of the textbook by Du and Ko.

Suggested Additional Reading:

• Two applications of inductive counting for complementation problems - Borodin, Cook, Dymond, Ruzzo, Tompa - SIAM J. Comput. Volume 18, Issue 3, pp. 559-578 (1989)
• Relativizations of the P vs NP Question. T. P. Baker, J. Gill, R. Solovay. - SIAM Journal on Computing, 4(4): 431-442 (1975)
• Relative to a Random Oracle A, P^A != NP^A != co-NP^A with Probability 1. C. H. Bennett, J. Gill. SIAM Journal on Computing, 10(1): 96-113 (1981)

Space Heirarchy Theorem, Time Heirarchy theorem, Non-determinism, Basic Complexity Classes P,NP,PSPACE,L,NL. Savitch's Theorem. PSPACE=NPSPACE.

References: Chapter 1 of the textbook by Du and Ko.

Suggested Additional Reading:

• Blum's Speed-up theorem : There are certain computable functions that have nothing at all like an optimal algorithm to compute them.
• Levin's Lower bound theorem : Every computable function does have a tight lower bound. (Notes by Eric Allender.)
• A New Look at Some Classical Results in Computational Complexity - Arnaldo Moura, Igor Carboni Oliveira

Contents : Introduction to the course, administrative details.
Recap of Turing machine model, Configurations. Notion of a resource, Blum axioms.
Tape reduction. Basic theorems for time and space, Linear Speedup theorem, Tape Compression Theorem.

References:

• Chapter 1 of Book on "Theory of Computational Complexity" - Du and Ko

Suggested Additional Reading:

• Machine-independent complexity theory [direct link, pdf] - Chapter in Handbook of Theoretical Computer Science (Vol. A): Algorithms and complexity - Joel I. Seiferas.
• Things that should have been taught in previous courses - Oded Goldreich.