Electronic Patient-Physician Communication: Problems and Promise
Promise Problems – Ilustrated
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H • AA • P
R • P
D • R
N • O
E • X
S • I
S • M
• A
O • T
F • I• O
• N
Goal:
• Show some approximation problems NP-hard
Plan:
• How to show inapproximability?• Probabilistically Checkable P
roofs (PCP)
• CLIQUE, COLORING
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Promise Problems – Ilustrated
*
• Must accept
Good inputs
• Must reject
Bad inputs
• Whatever
Other inputs
EG
Is there an assignment satisfying 90% of a
3SAT, assuming
one can satisfy 80%?
Gap problems? A special
case
• Best solution according to optimization parameter
Objective
Optimization
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• Find a solution within some factor of optimal
Approximation
problem
CLIQUE
3SAT
S.C.
min/max
max
max
min
instance
graph
3CNF
family
solution
clique
assign.
cover
parameter
|clique|
#clause
|cover|
optimal
algorithm
• C, efficient h-approximation for A resolves gap-A[C, hC]:
Observation:
<CYes
>hCNo
• Parameter is
In optimal Solution
GAP
Gap-A[C,hC] (minimization)
• Whatever
GAP
5accept if <hC
• 3-CNF
Instance:
• Maximum, over assignments, of fraction of satisfiable clauses:
Gap Problem: [7/8+, 1]
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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?
• E3CNF (clauses: exactly 3 independent literals) assignment that satisfies ≥7/8 of clauses
Claim:
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
• >0, Gap-3SAT[7/8+,1] is NP-hard
Theorem (PCP):
•Cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet
Proof:
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PCP Characterization of NPi.e.
LNP, Karp reduction f to
3CNF:xLf(x)3SAT
xLmax 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 -
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InputW
ork
Witness
for gap-3SAT: pick O(1) clauses
Instance:
• G=(V,E) (of m IS’s each of size 3)
Gap Problem:[(7/8+)/3 ,1/3]
• Parameter: fractional size of largest CLIQUE:
Yes
• ≥ 1/3
No
• < (7/8+)/310
Gap-CLIQUE
Theorem:
• Gap-CLIQUE [(7/8+)/3,1/3] 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
• Whatever11
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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Color CoordinationParty
• A set of partygoers; each pair of friends agree on possible pairs of colors
Solution: MAXV
• Invite some partygoers --- set colors s.t. all pairs OK
Optimize:
• How many invited
Solution: MAXE
• Assign a color to every partygoer
Optimize:
• Number of OK pairs
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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)
Instance:
• U=(V,E, [k], )
Gap Problem:[, ]
• Fractional size of largest all-consistent assignment:
Yes
• ≥
No
• < 15
gapV-kCSG[, ]
Observe:
• GapV-3CSG[(7/8+),1] is NP-hard
Theorem:
• >0 k=k()gap-IS[/k,1/k]NP-hard
How does one prove kCSG is NP-hard?Reduce from gap-3SAT?
3SAT is a ‘special case’
Proof: via kCSG
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CSGClaim [CSGIS]:
• 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’•I = (u, A(u))Yes
•A(u) = i | (u, i) INo
1 in each clique
u there is ≤1 such i
GapV-3CSG[(7/8+),1] is NP-hard
gapV-kCSG[, 1] L gapV-klCSG[l, 1]
gapV-kCSG[, 1] L gap-IS[/k, 1/k]
Whatever constraints, I can always invite 1% of partygoers
In that case, I can do 99%, whatever the constraints are!
How?
‘Invent’ a guest for every set of l real guests
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Party on
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AmplifyTheorem:
• gapV-kCSG[, 1] L gapV-klCSG[l, 1]
Proof:
• V’=Vl, ’=l
• E’: (u’, v’) s.t. uu’,vv’ s.t. u=v or (u, v)E
• ’ forbids: A(u’)u A(v’)u or (A(u’)u, A(v’)v)(u, v)
•Natural, consistent assignmentYes
•Color uu' by A(u’)u if any such u’V’ is colored. u’V’ is colored only if all uu’ are coloredNo
GapV-3CSG[(7/8+),1] is NP-hard
gapV-kCSG[, 1] L gapV-klCSG[l, 1]
gapV-kCSG[, 1] L gap-IS[/k, 1/k]
Constraints depend only on difference I can invite of partygoers
In that case, I can do the same for general constraints
And what does it get you?
Show approximate coloring of a graph is NP-hard
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Settle differences
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If colors are (mod q)
constraint (u, v) depends only on A(u)-A(v) (mod q)
Many symmetric colorings
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qCSGqCSG Instance:
• U = (V, E, , ) : EP[2] is define by
: E P{[q]} thus (u, v) = {(i,j) | i-j mod q (u, v)}
Theorem:
• gapV-kCSG[, 1] L gapV-qCSG[, 1]q = (nk)5
P{[q]} = all subsets of {0..q-1}
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q-ColoringCorollary:
• gap-[q, q/] is NP-hard
Proof:
• Apply same CSGIS reduction
• Consistent A d Ad(v)=A(v)+d (mod q) consistentYes
•vertexes partitioned into IS’s of fractional size /q – how many?No
q disjoint IS covering all
Lemma:• For q=|X|5, can efficiently
construct T:X[q] s.t. x1,x2,x3,x4,x5,x6,T(x1)+T(x2)+T(x3) T(x4)+T(x5)+T(x6) (mod q) {x1,x2,x3}={x4,x5,x6}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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Proof:
• q=(|V|||)5
• (u, v) = {T(u,i)-T(v,j) mod q | (i,j)(u,v)}
Pairs: assuming A’ colors vertexes u,v
• !d s.t. A’(u)=T(u,i)+d & A’(v)=T(v,j)+d for (i,j)(u,v)
Triplets: assuming A’ colors vertexes u,v,w
• dA’(u,v)=dA’(v,w) as A’(u)-A’(v) + A’(v)-A’(w) + A’(w)-A’(u)0 (mod q)
General: assuming A’ any partial coloring
• If dA’ is not everywhere the same, inconsistent triplet (u,v,w)
qCSG
denote: dA’(u,v) = that unique d
•for good A: A’(u) = T(u, A(u)) goodYes•for good A’: !d A(u)=T-1(u,A’(u)-d) goodNo
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