1
CS151
Complexity Theory
Lecture 16
May 19, 2011
May 19, 2011
2
New topic(s)
Optimization problems,
Approximation Algorithms,
and
Probabilistically Checkable Proofs
May 19, 2011
3
Approximation Algorithms
• “gap

producing” reduction
from
NP

complete problem L
1
to L
2
no
yes
L
1
L
2
(min. problem)
f
opt
k
rk
May 19, 2011
4
Gap producing reductions
•
rgapproducing reduction
:
–
f computable in poly time
–
x
L
1
opt(f(x))
k
–
x
L
1
opt(f(x))
> rk
–
for max. problems
use “
k
” and “
< k/r
”
•
Note: target problem is not a language
–
promise problem
(yes
no
not
all strings)
– “promise”: instances always from (yes
no)
May 19, 2011
5
MAXkSAT
•
Missing link:
first gapproducing reduction
– history’s guide
it should have something to do with SAT
•
Definition:
MAXkSAT with gap
ε
–
instance:
kCNF
φ
–
YES:
some assignment satisfies
all
clauses
–
NO:
no assignment satisfies more than (1
–
ε
)
fraction of clauses
May 19, 2011
6
Proof systems viewpoint
•
MAXkSAT with gap
ε
NP
hard
for any
language
L
NP
proof system of form:
–
given x, compute reduction to MAXkSAT:
x
–
expected proof is
satisfying assignment for
x
–
verifier picks
random clause
(“local test”) and
checks that it is satisfied by the assignment
x
L
Pr[verifier accepts] = 1
x
L
Pr[verifier accepts] ≤
(1
– ε)
–
can repeat O(1/
ε
) times for error < ½
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