CMPT 710 - Complexity Theory: Lecture 23
Valentine Kabanets
November 30, 2010
1
Barriers
Why cant we resolve the P vs. NP question? One of the basic answers is: We need new
techniques! But what are the old techniques and why are they not useful for resolv

CMPT 710 - Complexity Theory: Lecture 22
Valentine Kabanets
December 12, 2010
1
Gap-producing and gap-preserving reductions
How can we show that some minimization problem (such as VC) is NP-hard to approximate
to within some factor g? Well, we need to giv

CMPT 710 - Complexity Theory: Lecture 7
Valentine Kabanets
September 29, 2010
1
NP Completeness
Circuit-SAT = cfw_C | C is a satisable Boolean circuit
Theorem 1 (Cook-Levin). Circuit-SAT is NP-complete.
Proof. We need to prove that
1. Circuit-SAT is in NP

CMPT 710 - Complexity Theory: Lecture 19
Valentine Kabanets
November 16, 2010
1
Interactive Protocols
The interactive protocols (IP) are also protocols between a probabilistic polytime Verier and
an all-powerful Prover, where after a certain number of rou

CMPT 710 - Complexity Theory: Lecture 21
Valentine Kabanets
November 23, 2010
1
Generalization to the IP = PSPACE
Since TQBF is a complete problem for PSPACE, it suces to give an IP protocol for TQBF.
A QBF can be of the form x1 x2 x3 . . . (x1 , x2 , . .

CMPT 710 - Complexity Theory: Lecture 10
Valentine Kabanets
October 7, 2010
1
Dichotomies
Weve seen some examples of problems in NP, and most of these were either in P or NPcomplete. Can it be the world has such a simple structure that every NP-problem is

CMPT 710 - Complexity Theory: Lecture 4
Valentine Kabanets
September 21, 2010
1
Reductions: Example
3SAT = cfw_ | is a satisable 3CNF formula (Recall that a 3CNF is a conjunction
of clauses, where each clause is a disjunction of three literals; a literal

CMPT 710 - Complexity Theory: Lecture 17
Valentine Kabanets
November 5, 2010
1
BPP and P
It is a big open question whether every language in BPP can be decided in deterministic
polytime, i.e., whether BPP = P. There is some evidence that this equality ind

CMPT 710 - Complexity Theory: Lecture 2
Valentine Kabanets
September 9, 2010
1
Review
1. Problems and Languages
2. Turing machines
3. Reductions
4. Complexity classes
5. Completeness
1.1
Problems and Languages
A function problem is a function from strings

CMPT 710 - Complexity Theory: Lecture 8
Valentine Kabanets
September 30, 2010
SAT = cfw_ | is a satisable Boolean formula
Theorem 1 (Cook-Levin). SAT is NP-complete.
Proof. SAT is in NP (easy). To prove NP-hardness, we will show that Circuit-SAT is reduci

CMPT 710 - Complexity Theory: Lecture 16
Valentine Kabanets
November 3, 2010
1
BPP and Small Circuits
The ability to reduce the error probability in BPP has a curious consequence that every
language in BPP is computable by a family of polysize Boolean cir

CMPT 710 - Complexity Theory: Lecture 24
Valentine Kabanets
December 7, 2010
1
Natural proofs
The P vs. NP question is about uniform complexity classes. The conjectured inequality
between these two classes means that SAT is not computable by a uniform pol

CMPT 710 - Complexity Theory: Lecture 3
Valentine Kabanets
September 14, 2010
1
1.1
Turing machines (continued)
Example: PALINDROMES
Let PALINDROMES=cfw_w cfw_0, 1 | w = wR be the set of all binary palindromes (strings
that read the same forward and back

CMPT 710 - Complexity Theory: Lecture 5
Valentine Kabanets
September 22, 2010
1
Robust Time and Space Classes
Robust (intuitive notion): no reasonable changes to the model of computation should
change the class; capable of classifying interesting problems

CMPT 710 - Complexity Theory: Lecture 13
Valentine Kabanets
October 20, 2010
1
1.1
Small circuits vs. Collapse of PH
?
NP P/poly
There is an interesting connection between NP having polysize circuits and the collapse of a
polytime hierarchy.
Theorem 1 (Ka

CMPT 710 - Complexity Theory: Lecture 18
Valentine Kabanets
November 9, 2010
1
Randomized NP
Randomness is a computational resource that can be combined with other resources, e.g.,
nondeterminism. We will consider such a combination next.
Recall that a la

CMPT 710 - Complexity Theory: Lecture 6
Valentine Kabanets
September 23, 2010
1
Padding Technique
c
Suppose L EXP is decided by a TM M in time 2n , for some constant c. Dene a new
language
|x|c
Lpad = cfw_x#2 | x L
(here, # is a new symbol outside the alp

CMPT 710 - Complexity Theory: Lecture 20
Valentine Kabanets
November 18, 2010
1
Public-coin protocol for Graph NonIsomorphism
Theorem 1 (Goldwasser-Sipser). N ISO AM
Proof of Theorem 1. With loss of generality, assume that given input graphs (G1 , G2 ) ha

CMPT 710 - Complexity Theory: Lecture 14
Valentine Kabanets
October 22, 2010
1
1.1
Parallel Computation
NC
Imagine a Boolean formula on n variables. Suppose that we apply the appropriate electric
currents to the inputs. How long will take for these curren

CMPT 710 - Complexity Theory: Lecture 11
Valentine Kabanets
October 12, 2010
1
PSPACE-Completeness
Recall the NP-complete problem SAT: Is a given Boolean formula (x1 , . . . , xn ) satisable?
The same question can be stated equivalently as: Is the formula

CMPT 710 - Complexity Theory: Lecture 15
Valentine Kabanets
November 9, 2010
1
Randomized Computation
Why is randomness useful? Imagine you have a stack of bank notes, with very few counterfeit
ones. You want to choose a genuine bank note to pay at a stor