Last class

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Last class
Decision/Optimization 3-SAT? Independent-Set Indep
endent-Set ? 3-SAT P, NP Cooks Theorem NP-hard,
NP-complete 3-SAT ? Clique, Subset-Sum, 3-COL
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Reductions
A ? B
all reductions we had were
INSTANCE of A ? INSTANCE of B
(many-to-one reductions)
the black-box intuition model allowed more
questions to an oracle for B
(Turing reductions)
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Planar-3-COL
INSTANCE planar graph G QUESTION can the
vertices of G be assigned colors
red,green,blue so that no two neighboring
vertices have the same color?
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3-COL ? Planar-3-COL
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4-COL
INSTANCE graph G QUESTION can the
vertices of G be assigned one of 4 colors so
that no two neighboring vertices have the
same color?
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3-COL ? 4-COL
?
G
G
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planar 4-COL
INSTANCE planar graph G QUESTION can the
vertices of G be assigned one of 4 colors so
that no two neighboring vertices have the
same color?
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planar 3-COL ? planar 4-COL ?
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planar 3-COL ? planar 4-COL ?
planar 4-COL is very easy the answer is
always yes. (4-color theorem, Appel, Haken)
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Integer linear-programming
INSTANCE variables x1,...,xn
collection of linear inequalities over the xi
with integer coefficients QUESTION
does there exist an assignment of integers
to the xi such that all the linear inequalities
are satisfied?
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Integer linear-programming
INSTANCE variables x1,...,xn
collection of linear inequalities over the xi
with integer coefficients QUESTION
does there exist an assignment of integers
to the xi such that all the linear inequalities
are satisfied?
x1 ? 1 x2 ? 16 x1 x3 ? 16 x2 x4 ? 16 x3 x3x4x1
? 10000
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Integer linear-programming
we will show that ILP is NP-hard by showing
3-SAT ? ILP
true 1, false 0
y1 ? ? y2 ? y3 ? x1 (1-x2) x3 ? 1
0? x1? 1

.... 0?
xn? 1
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Integer linear-programming
Is integer linear programming NP-complete ?
I.e., is ILP in NP ?
Witness of solvability solution, but a priori
we do not know that the solution is polynomially
bounded.
ILP ? NP, but the proof is far from trivial.
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Min-Cut problem
cut S ? V number of edges
crossing the cut u,v u? S, v?
V-S
INPUT graph G OUTPUT cut S with the minimum
number of crossing edges
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Min-Cut problem
in P for each s,t pair run max-flow algorithm
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Max-Cut problem
cut S ? V number of edges
crossing the cut u,v u? S, v?
V-S
INPUT graph G OUTPUT cut S with the maximum
number of crossing edges
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Max-Cut problem
INSTANCE graph G, integer K QUESTION does G
have a cut with ? K
crossing edges?
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Max-Cut problem
INSTANCE graph G, integer K QUESTION does G
have a cut with ? K
crossing edges?
NAE-3-SAT ? Max-Cut
NAE-3-SAT INSTANCE 3-CNF formula
QUESTION does there exist an assignment such
that every claues have ? 1 false and ? 1 true ?
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NAE-3-SAT ? Max-Cut
INSTANCE 3-CNF formula QUESTION does there
exist an assignment such that every claues have
? 1 false and ? 1 true ?
1 vertex for each literal
x1 ? ? x2 ? x3
x3
x2
2m parallel edges
x1
? x2
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3-SAT ? NAE-3-SAT
y1 ? y2 ? y3 ? y1? y2? zi , ? zi ?
y3 ? b

• C satisfiable ? can find 3-NAE assignment for C
• C has 3-NAE assignment ? C satisfiable

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Is NP ? co-NP P ?
Factoring
INPUT integer n OUTPUT factorization of n,
i.e., np1?1 ... pk?k
Factoring decision version
INSTANCE pair of integers n,k QUESTION does n
have a factor x?2,...k ?