CS 535: Complexity Theory, Fall 2023

Course Overview

The goal of computational complexity theory is to understand the capabilities and fundamental limitations of efficient computation. In this course, we will ask questions such as, "What kinds of computational problems are inherently difficult?" in that solving them requires massive running times, no matter how cleverly one tries to design algorithms. In addition to running time, we will also quantify efficiency in terms of the use of other computational resources including space (memory), randomness, nondeterminism, interaction, communication, and algebraic operations. The learning objectives of the course are to:
Instructor:   Mark Bun, mbun [at] bu [dot] edu
Instr. Office Hours:   Mon 3-4:30 (CDS 1021)
    Thu 5-6 (CDS 1021)
Teaching Fellow:   Mandar Juvekar, mandarj [at] bu [dot] edu
TF Office Hours:   Tue 3:30-4:30 (CDS 1001)
    Fri 10:30-11:30 (CDS 614)
Class Times:   Tue, Thu 2:00-3:15 (MCS B31)
Discussion Section:   Wed 11:15-12:05 (BRB 121)

Important Links

Course Website: https://cs-people.bu.edu/mbun/courses/535_F23. The website contains the course syllabus, schedule with assigned readings, homework assignments, and other course materials.

Piazza: https://piazza.com/bu/fall2023/cs535. The entry code will be emailed to all registered students just prior to the start of the semester; please let the course staff know if you need it sent to you directly. All class announcements will be made through Piazza, so please set your notifications appropriately. Please post questions about the course material to Piazza instead of emailing the course staff directly. It is likely that other students will have the same questions as you and may be able to provide answers in a more timely fashion. Active participation on Piazza may add extra points to your participation grade.

Gradescope: https://gradescope.com. Sign up for a student account on Gradescope using your BU email address. The entry code for the course is ZWZGPW. Homework assignments are to be submitted to Gradescope in PDF format.

Catalog Description

Covers topics of current interest in the theory of computation chosen from computational models, games and hierarchies of problems, abstract complexity theory, informational complexity theory, time-space trade-offs, probabilistic computation, and recent work on particular combinatorial problems.


CS 332 (Theory of Computation) or a similar rigorous undergraduate introduction to the theory of computation. Aside from the formal prerequisite, it's important to have "mathematical maturity": Comfort with mathematical abstraction, a solid understanding of basic combinatorics and discrete probability, and the ability to read, understand, and write mathematical proofs. If you have not completed the prerequisites for the course, please schedule a meeting with me before registering.

List of potential topics

(Perpetually Tentative) Schedule

Date Topics Arora-Barak Reading Handouts/Assignments
Tue 9/5 Course welcome, Turing machines, decidability 0, 1.1-1.5, 1.7 (optional), A.1-A.2 L1 notes Collaboration Policy, Survey, hw1.pdf, hw1.tex, macros.tex
Thu 9/7 Time complexity, P, NP, NP-completeness 1.6, 2.1, 2.6-2.7 L2 notes
Tue 9/12 NP-completeness, Cook-Levin Theorem 2.2-2.4 L3 notes HW 1 due, hw2.pdf, hw2.tex
Thu 9/14 Decision vs. search 2.5 L4 notes
Tue 9/19 Hierarchy theorems, Ladner's Theorem 3.1-3.3, 4.1.3 L5 notes HW 2 due, hw3.pdf, hw3.tex
Thu 9/21 Relativization, Space complexity 3.4, 4.1 L6 notes
Tue 9/26 Savitch's Theorem, PSPACE, PSPACE-completeness 4.2 L7 notes HW 3 due, hw4.pdf, hw4.tex
Thu 9/28 Logspace computation 4.3 L8 notes
Tue 10/3 Immerman-Szelepcsényi Theorem, Polynomial hierarchy 4.3, 5.1-5.2 L9 notes HW 4 due, hw5.pdf, hw5.tex
Thu 10/5 PH via oracles, alternation 5.3, 5.5, L10 notes
Tue 10/10 NO CLASS (Virtual Monday)
Thu 10/12 More alternation, time-space tradeoffs 5.3-5.4, L11 notes
Tue 10/17 Circuits, non-uniform computation 6.1-6.3, L12 notes HW 5 due, hw6.pdf, hw6.tex
Thu 10/19 Karp-Lipton Theorem, circuit lower bounds, restricted circuit classes 6.4-6.7, L13 notes
Tue 10/24 Midterm Exam
Thu 10/26 Probabilistic algorithms A.2, 7.1-7.2, L14 notes
Tue 10/31 Randomized time classes, concentration inequalities A.2, 7.3-7.4, L15 notes HW 6 due, term paper topic due, hw7.pdf, hw7.tex
Thu 11/2 Error reduction, BPP vs. P/poly, BPP vs. PH 7.4-7.5, L16 notes
Tue 11/7 PromiseBPP, randomized reductions, Valiant-Vazirani Theorem 7.5, 17.4.1, L17 notes HW 7 due, hw8.pdf, hw8.tex
Thu 11/9 Counting, #P 17.1-17.3.1, L18 notes
Tue 11/14 #P-completeness 17.2-17.3, L19 notes HW 8 due, hw9.pdf, hw9.tex
Thu 11/16 Toda's Theorem, Interactive proofs 17.4, 8.1, L20 notes
Tue 11/21 Arthur-Merlin classes 8.2, L21 notes Term paper draft due
Thu 11/23 NO CLASS — Happy Thanksgiving!
Tue 11/28 IP = PSPACE 8.3, L22 notes
Thu 11/30 PCP Theorem, hardness of approximation 11.1-11.3, L23 notes Term paper feedback due
Tue 12/5 More hardness of approximation, proof of PCP Mini 11.4-11.5, L24 notes HW 9 due
Thu 12/7 Quantum circuits 10.1-10.3, L25 notes
Tue 12/12 Grover's algorithm 10.4, L26 notes Term paper due
Tue 12/19 Final Exam 3-5PM

Texts and References

Required Textbook

Sanjeev Arora and Boaz Barak, Computational Complexity: A Modern Approach. ISBN-13: 978-0521424264.

Other Resources

Oded Goldreich, Computational Complexity: A Conceptual Perspective.
Steven Homer and Alan L. Selman, Computability and Complexity Theory.
Cristopher Moore and Stephan Mertens, The Nature of Computation.
Christos H. Papadimitrou, Computational Complexity.
Michael Sipser, Introduction to the Theory of Computation.
Avi Wigderson, Mathematics and Computation.


Your grade in the course will be determined by homework assignments, three take-home tests, a term paper, and class participation.

Homework (50%)

There will be weekly homework assignments due each Tuesday at 11:59PM. Homework sets are designed to be challenging, so you will want to start early to give yourself time to think deeply about the problems. No late homework will be accepted. To accomodate extenuating circumstances, your lowest homework grade will be dropped. Further accomodations require a note from your academic advisor.

You are allowed, and indeed encouraged, to collaborate with other students on solving the homework problems. However, you must write the solutions independently in your own words. Details of the collaboration policy may be found here: Collaboration and Honesty Policy

Some homework assignments will include clearly marked "individual review" problems. These problems are meant to reinforce concepts from previous assignments that you've already received feedback on. These problems are to be completed individually; no collaboration is allowed.

Some homework assigments may also include optional "bonus" problems. Solving these problems will not directly contribute to your homework grade but may improve the letter grade you receive in the course if the final percentage we calculate is on the borderline between two letter grades. Solving bonus problems is also a good way to impress your instructor if you are seeking a recommendation letter, research opportunities, or a grading position.

Homework solutions must be typeset. LaTeX is the standard document preparation system used in the mathematical sciences, but you are also free to use other tools such as Microsoft Word. If you wish to include drawings or figures, you may draw them by hand and incorporate the images into your documents. (There are packages for creating images within LaTeX, but they can be unnecessarily time-consuming to use.)

My preferred LaTeX editors are TexShop for Mac and TexStudio for Windows. If you would like to give LaTeX a try on the web without installing anything on your computer, Overleaf is a good option.

Not so short intro to LaTeX. A LaTeX tutorial.

For your convenience, we will supply the LaTeX source for each assignment along with the compiled PDF. Changing the flag \inclsolns from 0 to 1 will let you add your name, collaborators, and solutions directly to the assignment. The file macros.tex should be in the same directory for it to compile correctly.

In-Class Tests (25%)

There will an in-class midterm on Tuesday, October 24. A comprehensive in-class final will be held during our registrar-appointed two-hour exam slot. Please wait until the official University final exam schedule is finalized before making your end-of-semester travel plans.

You may bring one double-sided 8.5" x 11" sheet of notes to the midterm test and two such sheets to the final. Note sheets may be either handwritten or typeset. You may not use any other aids during the exam, including but not limited to books, lecture notes, calculators, phones, or laptops.

Term Paper (15%)

At the end of the course, you will complete a final mini-project that involves writing a 5-10 page review of a research article or survey in complexity theory. Your review should be targeted to other students in the class. The project will also involve supplying constructive feedback to your fellow students. Term papers may be written individually or in pairs. Details for the assignment are below.

Term Paper Instructions

Class Participation (10%)

Thoughtfully engaging with the readings, in lectures, and in discussion sections are all great ways to build your understanding of the course material. We'll keep track of your participation in these activities in two main ways.

Reading Responses: To help keep you on track with reading assignments, we ask you to submit brief reading responses to Piazza every week. A reading response consists of at least one insightful question or comment about the week's assigned readings. (More are encouraged!) Reading responses are due by 10PM every Sunday, but submissions received before the relevant class period will help us use our time in class together to work through the most important and difficult parts of the material.

Here are some suggestions to guide your thinking as you comment on the reading.

If you are not comfortable posting some questions or comments publicly, you may set their visibility to "instructors only."

(These guidelines are adapted from Salil Vadhan's CS221 and other courses.)

Discussion Sections: We will record active attendance in our weekly discussion sections, but will not grade your solutions to discussion problems.

You can earn full participation credit by completing nine (9) reading responses and attending nine (9) discussion sections. Your participation score can be also supplemented by thoughtfully asking and answering questions in lectures, in discussions, on Piazza, or during office hours.

Course Policies

Academic Conduct

All Boston University students are expected to maintain high standards of academic honesty and integrity. It is your responsibility to be familiar with the Academic Conduct Code, which describes the ethical standards to which BU students are expected to adhere and students’ rights and responsibilities as members of BU’s learning community. All instances of cheating, plagiarism, and other forms of academic misconduct will be addressed in accordance with this policy. Penalties for academic misconduct can range from failing an assignment or course to suspension or expulsion from the university.


Attendance Policy

Students are expected to attend each class session unless they have a valid reason for being absent. If you must miss class due to illness or another reason, please notify the instructor as soon as possible, ideally before the absence.


Absence Due to Religious Observance

If you must miss class due to religious observance, you will not be penalized for that absence and you will receive a reasonable opportunity to make up any work or examinations that you may miss. Please notify the instructor of absences for religious observance as soon as possible, ideally before the absence.



In the event of the death of an immediate family member, you should notify your advisor, who will help you coordinate your leave. You will be automatically granted five weekdays of leave, and if necessary, you advisor will help you to petition the Dean for additional leave time. You may also request a leave of absence due to bereavement. Please contact your advisor, who will help you with the process.


Disability Services

Students with documented disabilities, including learning disabilities, may be entitled to accommodations intended to ensure that they have integrated and equal access to the academic, social, cultural, and recreational programs the university offers. Accommodations may include, but are not limited to, additional time on tests, staggered homework assignments, note-taking assistance. If you believe you should receive accommodations, please contact the Office of Disability Services to discuss your situation. This office can give you a letter that you can share with instructors of your classes outlining the accommodations you should receive. The letter will not contain any information about the reason for the accommodations.

If you already have a letter of accommodation, you are encouraged to share it with your instructor as soon as possible.

Disability & Access Services
25 Buick Street, Suite 300

Grade Grievances

If you feel that you have received an arbitrary grade in a course, you should attempt to meet with the grader before filing a formal appeal. If the student and the instructor are unable to arrive at a mutually agreeable solution, the student may file a formal appeal with the chair. This process must begin within six weeks of the grade posting. To understand how an “arbitrary grade” is defined, please explore the following link.


Incomplete Grades

An incomplete grade (I) is used only when the student has conferred with the instructor prior to the submission of grades and offered acceptable reasons for the incomplete work. If you wish to take an incomplete in this class, please contact the instructor as soon as possible but certainly before the submission of final grades. To receive an incomplete, you and your instructor must both sign an “Incomplete Grade Report” specifying the terms under which you will complete the class.


Student Health Services

Offers an array of health services to students, including wellness education and mental health services (behavioral medicine).


Medical Leave of Absence

If you must take a leave of absence for medical reasons and are seeking to re-enroll, documentation must be provided to Student Health Services so that you may re-enroll. To take a medical leave, please talk with SHS and your advisor, so that they may assist you in taking the best course of action for a successful return.



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