University of Toronto -- Department of Computer Science
St. George Campus -- Fall Session 2004

CSC 363H: "Computational Complexity and Computability"

Lecture Summaries and Tutorial Exercises

Contents

• Learning objectives.
• Outline of lecture topics.
• Lecture summaries.
• Tutorial exercises.

The schedule of lectures, tutorials, and office hours, is available on the main page.

| Top | Contents | Objectives | Outline | Lectures | Tutorials | Bottom |

Learning objectives

By the end of this course, students should

• be familiar with fundamental restrictions on computability:

  • be familiar with Turing machines;
  • understand the notions of decidability and semi-decidability;
  • know that the Halting problem is undecidable;
  • understand and be able to apply techniques for showing undecidability (diagonalization, reductions);

• understand the importance of computational complexity:

  • be familiar with the definitions of P and NP;
  • be able to demonstrate membership in P or NP;
  • be familiar with the definition of NP-completeness;
  • be able to apply techniques for showing NP-completeness;
  • know heuristics and approximation algorithms for dealing with NP-complete problems;
  • know standard lower bounds for common problems.

| Top | Contents | Objectives | Outline | Lectures | Tutorials | Bottom |

Outline of lecture topics

The following topics will be covered in this course, in the order listed. For each topic, we have indicated the approximate number of weeks required to cover that topic as well as a list of the relevant sections in the textbook.

• Computability Theory [5 weeks]
  (Chapters 3, 4, 5 in the textbook)
  • Turing machines: definitions and examples [1 week] (section 3.1)
  • Other models of computation, the Church-Turing thesis [1 week] (sections 3.2, 3.3)
  • Diagonalization, the Halting problem, recognizability [1 week] (sections 4.1, 4.2)
  • Reducibility, examples [1 week] (sections 5.1, 5.2)
  • Mapping reducibility, examples [1 week] (section 5.3)
• Complexity Theory [8 weeks]
  (Chapters 7, 10 in the textbook)
  • Models of efficient computation
  • P and NP
  • NP-completeness, Cook's theorem
  • Self-reducibility and polytime transformations
  • Approximation algorithms and heuristics
  • Lower bounds on problem complexity (information-theory bounds, adversarial arguments).

| Top | Contents | Objectives | Outline | Lectures | Tutorials | Bottom |

Lecture summaries

Every week, specific sections of the textbook will be posted as readings. You will be expected to read these sections to prepare for the following week's lectures and tutorials.
At the end of each week, a short summary of the material covered during lecture will be posted.
All summaries will be plain text (ASCII) files, unless otherwise indicated.

1. Week 1: summary;
   readings: 3.1 (and review chapters 0, 1, 2 briefly).
2. Week 2: summary;
   readings: 3.2, 3.3.
3. Week 3: summary;
   readings: re-read end of Chapter 3, 4.1 except for "Decidable problems concerning context-free languages", 4.2 except for "The diagonalization method".
4. Week 4: summary;
   readings: "The diagonalization method" in 4.2 (pp.160-164), 5.1.
5. Week 5: summary;
   readings: 5.3.
6. Week 6: summary;
   readings: 5.3, 7.1.
7. Week 7: summary;
   readings: 7.2, 7.3.
8. Week 8: summary;
   readings: end of 7.3 ("examples", "P vs NP"), start of 7.4 ("polytime reducibility", "NP-completeness").
9. Week 9: summary;
   readings: end of 7.4 ("Cook-Levin theorem"), 7.5.
10. Week 10: summary updated on Fri 26 Nov;
    readings: quickly look at 8.1, 8.2, 8.3, 9.1 (read just the results of the theorems, not the proofs).
11. Week 11: summary;
    readings: 10.1.
12. Week 12: summary;
    readings: 10.1.
13. Week 13: summary;
    finish approximation algorithms and heuristics; review; discussion of exam (check out Tests/Exam page).

| Top | Contents | Objectives | Outline | Lectures | Tutorials | Bottom |

Tutorial exercises

At the end of every week, a set of tutorial exercises will be posted for the following week. You will be expected to work on these exercises (either alone or in group) and to bring your solution ideas to tutorial, where your TA will discuss them with you.
All exercises will be plain text (ASCII) files, unless otherwise indicated.

1. Week 1: no tutorial
2. Week 2
3. Week 3 Updated on Mon 20 Sep 2004.
4. Week 4 Updated on Tue 28 Sep 2004.
5. Week 5
6. Week 6: Term Test 1
7. Week 7
8. Week 8
9. Week 9
10. Week 10
11. Week 11
12. Week 12: Term Test 2
13. Week 13

| Top | Contents | Objectives | Outline | Lectures | Tutorials | Bottom |