6.841 Advanced Complexity Theory
Feb 4, 2009
Lecture 1
Lecturer: Madhu Sudan
Scribe: Mergen Nachin
1
Administrative Information
•
Lecturer: Madhu Sudan ([email protected])
•
TA: Brenden Juba ([email protected])
•
Website:
http://courses.csail.mit.edu/6.841/
The grading will be based on the following.
•
Scribing  You must scribe at least one lecture no matter if you are taking the class for credit or as a
listener.
•
Problem sets  There will be roughly 3 problem sets throughout the semester.
•
Participation  We encourage people to speak up, discuss and ask questions during the lecture.
•
Project  Read papers about some topic and present it to the class (with additional progress, if possible).
2
High level overview of Computational Complexity
Computational Complexity is concerned with the study of
•
Interesting
computational problems.
•
Interesting
resources such as time, space and etc.
•
The feasibility and infeasibility  That is to prove upper and lower bounds. Unfortunately, we have
a very few results on lower bounds for time or space.
But on the other hand, we have made quite
a progress on
comparison
lower bounds.
For example, we compare two problems and conclude the
following: if problem A requires some certain amount of resource to solve, then we must need at least
some amount of resource to solve problem B.
How do we define “interesting”?
This is a very subjective choice.
For example, a problem might be
interesting if it has a lot of applications in a real world, or if many other problems can be reduced to one of
its instances. Once we find an “interesting” problem, we want to find out how much time and space suffice
to solve the problem, and how much are necessary to solve the problem.
2.1
Examples of “interesting” problems
The following three problems are presented as “interesting”.
•
#SAT (“numberSAT”): Given a 3CNF formula
φ
on
n
variables
x
1
, . . . , x
n
with
m
clauses
c
1
, . . . , c
m
(so
φ
=
c
1
∧
. . .
∧
c
m
and each
c
i
looks something like
x
i
1
∨
¯
x
i
2
∨
x
i
3
), count the number of satisfying
assignments.
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 Spring '09
 MadhuSudan
 Computational complexity theory, Karp, Computational Complexity

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