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H •AA •PR •PD •RN •OE •XS •IS •M•AO •TF •I•O•N

Goal: •Show some approximation problems NP-hard

Plan:•How to show inapproximability?•Probabilistically Checkable Proofs (PCP)•CLIQUE, COLORING

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3

Promise Problems – Ilustrated

*

Good inputs

• Must accept

Bad inputs

• Must reject

Other inputs

• Whatever

EGIs there an assignment satisfying 90% of a 3SAT, assuming one can satisfy 80%?

Gap problems? A special case

Objective

• Best solution according to optimization parameter

Optimization

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Approximation

• Find a solution within some factor of optimal

problem

CLIQUE

3SAT

S.C.

min/max

max

max

min

instance

graph

3CNF

family

solution

clique

assign.

cover

parameter

|clique|

#clause

|cover|

optimal

algorithm

Observation:

• C, efficient h-approximation for A resolves gap-A[C, hC]:

<CYes

>hCNo

In optimal Solution

• Parameter isGAP

Gap-A[C,hC] (minimization)

GAP

• Whatever

5accept if <hC

Instance:

• 3-CNF

Gap Problem: [7/8+, 1]

• Maximum, over assignments, of fraction of satisfiable clauses:

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Gap-3SAT

Yes

• =1

No

• < 7/8+

EG( x1 x2 x3 )( x1 x2

x2 )( x1 x2

x3 )( x1 x2

x2 )(x1 x2 x3 )( x3 x3

x3 )

x1 F ; x2 T ; x3 F satisfies 5/6 of clauses

Why 7/8?

Claim:

• 3CNF (with exactly 3 independent literals), assignment that satisfies ≥7/8 of clauses

Proof:

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7/8

•How many does an assignment satisfy expectedly?E•clause Ci, let Yi be a 0/1 variable: “is Ci satisfied?” Yi

•Yi’s expectation: E[Yi] = 7/8E Yi

•E[ Yi] = E[Yi] = m7/8 (m = #clauses)E•assignment with at least that number satisfied

Theorem (PCP):

• >0, Gap-3SAT[7/8+,1] is NP-hard

Proof:

•Cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet

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PCP Characterization of NP

i.e.LNP, Karp reduction to 3CNF:xLf(x)3SATxLmax satisfy <7/8+ of f(x)

Approximating max-3SAT to within which factor is thus proven NP-hard?

Verify witness

• Choose only O(1) bits to read (randomly)

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Probabilistic Checking of Proofs

_ _

_ _

_ -

_ _

_ _

a a

b a

b -

-

InputW

ork

Witness

for gap-3SAT: pick O(1) clauses

The end of a fairy tale?

…now it’s all about money…

How can One ensure their money is safe in the bank?

While remaining unidentified!

Enter: a trusted banker

How would we

know you’re you?

Here is a problem

only I can solve

Is there some faster way?

Care if it’s rando

m?

Instance:

• G=(V,E) comprising of m independent sets of size 3

Gap Problem:[(7/8+)m,m]

• Depending on the size of largest CLIQUE:

Yes

• ≥ m

No

• < (7/8+)m12

Theorem:

• Gap-CLIQUE [(7/8+)m,m] is NP-hard

Corollary:

• It is NP-hard to approximate MAX-CLIQUE to within 7/8+

How does one prove such an NP-hardness theorem?Reduce from gap-3SAT?

Gap Preserving Reduction

*

*

f

GAP

• Whatever13

3SAT

CLIQUE

G,m

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SAT p CLIQUE

1 vertex for 1 occurrence• within triplets. =.inconsistency non-edge

m= number of clausesA clique of size l an assignment satisfying ≥l clauses

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Constraints` GraphCSG Input:

• U = (V, E, , ) - : EP[2]

Solution: MAXV

• A: V{} s.t. (u,v)E & A(u),A(v) (A(u),A(v)) (u,v)

Optimize:

• Pr[A(u)]

Solution: MAXE

• A: V

Optimize:

• Pr[(A(u),A(v))(u,v)]

EGCLIQUEIS3SATV.C.(G)

EGMAX-CUT(G)

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CSGObservation:

• gapV-3CSG[7/8+, 1] is NP-hard

Claim:

• gapV-kCSG[, 1] L gap-IS[/k, 1/k]

Proof:

• V’=V,• ij ((u,i),(u, j))E’

(i, j)(u, v)} ((u,i),(v, j))E’

special case 3SAT

•I = (u, A(u))Yes•A(u) = i | (u, i) INo

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CSG

You’d embezzle in another branchCan I send only very little info?

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CSGTheorem:

• gapV-kCSG[, 1] L gapV-klCSG[l, 1]

Proof:

• V’=Vl, ’=l

• E’,’ disallow all pairs with different colors to same vertex, or colors that dissatisfy any constraint

•Natural, consistent assignmentYes

•All occurring colors: colors of V to avoid 1- must be limited to lNo

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Gap-IS

Gap-IS[/k, 1/k]

NP-hard

>0

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What if colors are (mod q)

And all constraints depend only on the

difference

Many symmetric colorings

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qCSGqCSG Instance:

• : E [q] (u, v) = {(i,j) | i-j mod q (u, v)}

Theorem:

• gapV-kCSG[, 1] L gapV-(nk)5CSG[, 1]

Corollary:

• gap-[q, q/] is NP-hard

•Consistent A d Ad(v)=A(v)+d mod q consistent tooYes•vertexes partitioned into IS’s of fractional size /q – how many?No

q disjoint IS covering all

Lemma:

• For q=|U|5, can efficiently construct T:U[q] s.t. u1,u2,u3,u4,u5,u6,T(v1)+T(v2)+T(v3) T(v4)+T(v5)+T(v6) (mod q) {v1,v2,v3}={v4,v5,v6}

Proof:

• Incrementally assign values to members of U

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Triplets` Unique T

Corollary:

• T is unique for pairs and for single vertexes as well

Proof:

• T(x)+T(u)T(v)+T(w) (mod q) {x,u}={v,w}

• T(x)T(y) (mod q) x=y

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qCSGProof:

• q=(|V|||)5

• (u, v) = {T(u,i)-T(v,j) mod q | (i,j)(u,v)}

qCSG Instance:

• : E [q] (u, v) = {(i,j) | i-j mod q (u, v)}

Theorem:

• gapV-kCSG[, 1] L gapV-(nk)5CSG[, 1]

Pairs

(u,v) ! d s.t. A’(u)=T(u,i)+d, A’(v)=T(v,j)+d & (i,j)(u,v)

Let dA’(u,v) = that unique d

Tri

plets

u,v,w dA’(u,v) = dA’(v,w)as A’(u)-A’(v) + A’(v)-A’(w) + A’(w)-A’(u) = 0

General

If dA’ is not everywhere the same inconsistent triplet (u,v,w)

•A’(u) = T(u, A(u))Yes•d A(u)=T-1(u,A’(u)-d)No

Assume a complete graph – add trivial constraints

Question:

• Is it in P or NP-hard?

Constraints

• Colors` permutations

In optimal Solution

• Pr[(A(u),A(v))(u,v)]<No

>1-Yes

GAP

Gap-UG[, 1-] .

GAP

• Whatever

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Note:

• gap-Max-Cut[1-1.01, 1-] NP-hard

<1-No

>1-Yes

Optimal cut

• Pr[A(u)A(v)]GAP

Gap-MC[1-, 1-] .

GAP

• Whatever

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WWindex

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PCP PCP Theorem

Clique SAT

3SAT Vertex Cover

Coloring

CNF

NP-Hard

Interactive Proof System