1
CS151
Complexity Theory
Lecture 6
April 14, 2011
April 14, 2011
2
Clique
CLIQUE = { (G, k)  G is a graph with a
clique of size ≥ k }
(clique = set of vertices every pair of which are
connected by an edge)
•
CLIQUE is
NP
complete.
April 14, 2011
3
Circuit lower bounds
•
We think that
NP
requires exponentialsize
circuits.
•
Where should we look for a problem to
attempt to prove this?
•
Intuition: “hardest problems” –
i.e.,
NP

complete problems
April 14, 2011
4
Circuit lower bounds
•
Formally:
–
if
any
problem in
NP
requires
super
polynomial size circuits
–
then
every
NP
complete problem requires
superpolynomial size circuits
–
Proof idea
: polytime reductions can be
performed by polysize circuits using a variant
of CVAL construction
April 14, 2011
5
Monotone problems
•
Definition:
monotone language
= language
L
{0,1}
*
such that x
L implies
x’
L for all x
¹
x’.
–
flipping a bit of the input from 0 to 1 can only
change the output from “no” to “yes”
(or not at all)
April 14, 2011
6
Monotone problems
•
some
NP
complete languages are
monotone
–
e.g. CLIQUE (given as adjacency matrix):
–
others:
HAMILTON CYCLE, SET COVER
…
–
but not
SAT, KNAPSACK
…
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