A course on computational complexity for 4th year students (1st year of MSc programme).

Rules

Your grade will consist of:

• 10% quizes. Every lecture will have a quiz on moodle with several yes/no questions to check if you are paying attention.
• 40% homeworks. There will be 4 homeworks, some programming and some theory.
• 50% oral exam. At the end, there will be an oral exam.

Notice that there is no written exam, midterm or final. That is why the homeworks are bigger than usual. The exact point thresholds for the grades will appear later, but I promise that 50% (or less) will give you a passing grade (dostateczny).

The oral exams will be in the exercise groups. You can negotiate the date of the oral exam with your exercise teacher (ćwiczeniowiec), in justified cases it is possible to have the oral exam at an earlier date.

I expect that you, as a smart student, will:

Lectures

The lectures use a slide system that I am developing, so please send me bug reports! These lectures are roughly once per week, but some lectures are longer so there is no exact correspondence.

Lecture 1. Turing machines (October 4)

• Definition of Turing machines
• Decidability and semi-decidability
• Undecidability of the halting problem

Questions for the oral exam:

• prove that the halting problem is undecidable

Lecture 2. Space and time (October 11 and 18)

• The basic complexity classes: NL ⊆ P ⊆ NP ⊆ PSPACE ⊆ EXPTIME,
• separation of P and EXPTIME
• PSPACE = NPSPACE (Savitch)
• machines in finite space that always halt (Sipser Theorem)
• NL = coNL (Immerman and Szelepcsenyi Theorem)

Questions for the oral exam:

• Prove that P < EXPTIME
• Prove that PSPACE = NPSPACE
• Prove the Sipser Theorem
• Prove the Immerman and Szelepcsenyi Theorem

Lecture 3. Reductions and complete problems (October 25 November 8, and November 15)

• Turing and Karp reductions
• Sat as an NP-complete problem
• QBF as a PSPACE-complete problem
• alternating reachability as P-complete problem
• the Baker-Gill-Solovay Theorem.

Questions for the oral exam:

• Explain the difference between Turing and Karp reductions
• Prove that Sat is NP-complete, and that QBF is PSPACE-complete
• Prove the Baker-Gill-Solovay Theorem

Lecture 4. Introduction to circuits (November 22)

• Circuits and their “parallel algorithm” intuition
• Circuit descriptions of P and P/p0ly
• AC and NC

Questions for the oral exam:

• Define the class P/Poly and show that it is equal to polynomial size circuits

Lecture 5. Parity not in AC0 (November 22 and 29)

• parity is not in AC

Questions for the oral exam:

• (Only if you want grade 5) Prove that parity is not in AC

Lecture 6. Randomisation (December 6)

• the class RP ⊆ BPP ⊆ PP
• examples of RP algorithms, including polynomial identity testing and perfect matchings
• PP contains NP

Questions for the oral exam:

• Define the classes RP, BPP and PP
• Give RP algorithms for: evaluating straight line programs and polynomial identity testing

Lecture 7. Derandomisation (December 13)

• two techniques for derandomisation illustrated on max cut
• Adleman’s Theorem on BPP ⊆ P/Poly

Questions for the oral exam:

• Explain the two derandomised algorithms for max cut
• Prove Adleman’s Theorem

Lecture 8. Fine-grained complexity (December 20)

• SETH ⟹ Orthogonal vectors conjecture
• Orthogonal vectors conjecture  ⟹ NFA evaluation needs quadratic algorithms

Questions for the oral exam:

• Prove: SETH ⟹ Orthogonal vectors conjecture
• Prove: Orthogonal vectors conjecture  ⟹ NFA evaluation needs quadratic algorithms

Lecture 9. Parametrized complexity (January 10)

• An algorithm for vertex cover that is linear when the size of the vertex cover is fixed
• The class FPT and FPT-reductions
• An FPT algorithm for 3-colourability on bounded treewidth

Questions for the oral exam:

• Define the class FPT and show that vertex cover is in it
• Show FPT reductions, both ways, between the k-clique problem and Turing machines with accepting computations of length k.
• Prove that 3-colourability is FPT, where the parameter is treewidth

The remaining lectures are not part of the oral exam.

Lecture 10. Quantum computation (January 17)

• BPP via probabilistic circuits
• quantum circuits
• the class BQP (bounded error quantum polynomial time)

Lecture 11. Simon’s algorithm (January 24)

• Simon’s algorithm

Homework

There will be several homework assignments, including programming assignments.