[class poster]
[Games, Puzzles, and Computation book cover]

6.892 Algorithmic Lower Bounds: Fun with Hardness Proofs (Spring 2019)

Prof. Erik Demaine       TAs: Jeffrey Bosboom, Jayson Lynch

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Need to figure out when to give up the search for efficient algorithms?
Want to know why Tetris and Mario are computationally intractable?
Love seeing the connections between problems and how they can be transformed into each other?
Like solving puzzles that can turn into publishable papers?

This class takes a practical approach to proving problems can't be solved efficiently (in polynomial time and assuming standard complexity-theoretic assumptions like P ≠ NP). We focus on reductions and techniques for proving problems are computationally hard for a variety of complexity classes. Along the way, we'll create many interesting gadgets, learn many hardness proof styles, explore the connection between games and computation, survey several important problems and complexity classes, and crush hopes and dreams (for fast optimal solutions).

The ability to show a problem is computationally hard is a valuable tool for any algorithms designer to have. Lower bounds can tell us when we need to turn to weaker goals or stronger models of computation, or to change the problem we're trying to solve. Trying to find lower bounds can help us see what makes a problem difficult or what patterns we might be able to exploit in an algorithm. The hardness perspective can help us understand what makes problems easy, or difficult to solve.

Inverted Lectures

This year, we're experimenting with inverted lectures: most material is covered in video lectures recorded in 2014 (already watched by over 14,000 people), which you can conveniently play at faster speed than real time. In-class time will be focused on in-class problem solving, with some new material presented by the professor and/or guest lecturers. Particularly unusual is that the problems we'll solve in groups will include a choose-your-own-mix of problem-set style problems with known solutions, coding problems for those who love programming, and open research problems that no one knows the answer to, with the goal of publishing papers about whatever we discover. (The past offering of this class led to over a dozen published papers.) You can work on whatever type of problem most interests you. To facilitate collaboration, we'll be using a new open-source software platform called Coauthor, along with Github for (optional) coding.


This is an advanced class on algorithmic reduction. We will focus on techniques for proving problems are complete with respect to various complexity classes, not on the complexity theory itself. Here is a tentative list of topics:

Readings and Resources

There is no textbook for this class, but there are two recommended books, one book chapter, and several useful websites.
  1. Computers and Intractability A Guide to the Theory of NP-Completeness: book by Michael R. Garey and David S. Johnson
  2. Johnson's followup NP-completeness Columns
  3. Games, Puzzles, & Computation: book by Robert A. Hearn and Erik D. Demaineavailable at MIT from CRCnetBASE
  4. The Design of Approximation Algorithms by David Williamson and David Shmoys: chapter 16 is about inapproximability — ebook avalible from the MIT Library,     draft available for free download
  5. Limits to Parallel Computation: P-Completeness Theory by Raymond Greenlaw, H. James Hoover, and Walter L. Ruzzo — available for free download
  6. Complexity Zoo
  7. A compendium of NP optimization problems
  8. Erik's 6.046 notes proving some basic NP-hardness (3-Dimensional Matching, Subset Sum, 4-Partition)


Class Time: Wednesdays at 7:00pm–9:30pm
Class Room: MIT room 32-082
First Class: Wednesday, February 6, 2019
Bonus Time: Thursdays at 7:00pm–9:00pm   (optional)
Bonus Room: MIT room 32-082
Professor: Erik Demaine, 32-G680
TAs: Jeffrey Bosboom, 32-G632
  Jayson Lynch, 32-G580
Staff Email: 6892-staff at csail.mit.edu
Units: 3-0-9
Prerequisites: 6.006 or equivalent background in discrete mathematics and algorithms
Alternatively, permission from the instructor.
No background in computational complexity is required (though it doesn't hurt); all needed notions will be defined in this class.
Credit: Graduate, Independent Inquiry or AAGS (Theoretical Computer Science concentration).


We welcome both undergraduate and graduate students from all universities, although officially this is a graduate class.

If you are interested in attending the class, for credit or as a listener, please do the following:

  1. Join the 6892-students mailing list.
  2. Sign up for an account on Coauthor
  3. Fill out this signup form


There are five requirements for the class:

Past and Future

The class is offered only occasionally. The latest edition is Fall 2023. It was given in Fall 2014 as 6.890.