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PCPs and Inapproxiability CIS 6930
October 5, 2009
Lecture Hardness of Set Cover
Lecturer: Dr. My T. Thai
Scribe: Ying Xuan
1
Preliminaries
1.1
TwoProverOneRound Proof System
A new PCP model
2P1R
– Think of the proof system as a game between two provers
and one veriﬁer.
•
Prover Model:
–
each tries to cheat – convince the veriﬁer that a ”No” instance is ”Yes”;
–
each cannot communicate with the other on their answers;
•
Veriﬁer Model:
–
tries not to be cheated – ensures the probability to be cheated is upper
bounded by some small constant;
–
can crosscheck the two provers’ answers;
–
is only allowed to query
one
position in each of the two proofs.
Deﬁnition 1
Given three parameters: completeness
c
, soundness
s
and the number of
random bits provided to the veriﬁer
r
(
n
)
, assume the two proofs are written in two
alphabets
Σ
1
and
Σ
2
respectively, then a language
L
is in
2P1R
c,s
(
r
(
n
))
if there is
a polynomial time bounded veriﬁer
V
that receives
O
(
r
(
n
))
truly random bits and
satisﬁes:
•
for every input
x
∈
L
,
there is
a pair of proofs
y
1
∈
Σ
*
1
and
y
2
∈
Σ
*
2
that makes
V
accept with probability
≥
c
;
•
for every input
x /
∈
L
and
every pair
of proofs
y
1
∈
Σ
*
1
and
y
2
∈
Σ
*
2
makes
V
accept with probability
< s
;
2P1R
is actually a mechanism to solve LabelCover:
L
(
G
= (
U,V
;
E
)
,
Σ
,
Π)
⇔
u
∈
U
and
v
∈
V
, we have
•
v
: proof
P
1
;
u
: proof
P
2
;
•
uv
∈
E
:
u
,
v
have answers for the same bit;
•
a labeling
L
is said to satisfy an edge
uv
iff
Π
uv
(
L
(
u
)) =
L
(
v
)
: the two answers
are consistent, otherwise they are cheating.
Hardness of Set Cover1
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View Full DocumentYou can see this from the next proof and the proof of NPhard for GapMaxLabel
Cover
Σ
1
,η
in previous lecture.
Theorem 2 (29.21)
There is a constant
ϵ
p
>
0
such that
NP
=
2P1R
1
,
1

ϵ
p
(log(
n
))
.
Proof:
•
2P1R
1
,
1

ϵ
p
(log(
n
))
⊆
NP
(exercise);
•
NP
⊆
2P1R
1
,
1

ϵ
p
(log(
n
))
(shown here).
Given a SAT formula
ϕ
, we employ a
gapproducing
reduction to obtain a MAX
E3SAT(E5) instance
ψ
, that is, given
m
is the number of clauses in
ψ
:
•
if
ϕ
is satisﬁable,
OPT
(
ψ
) =
m
;
•
if
ϕ
is not satisﬁable,
OPT
(
ψ
)
<
1

ϵ
p
for some constant
ϵ
p
Goal
: Prove
SAT
∈
2P1R
1
,
1

ϵ
p
(log(
n
))
Protocol:
•
V
selects an index of a clause, sends it to the ﬁrst prover, selects a random
variable in the clause, and sends it to the second prover.
•
First prover returns 3 bits (the assignment of the selected clause), second returns
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