NPcomplete examples

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NP-complete examples
Fall 2009

• CSC3130 Tutorial 11

Xiao Linfu
lfxiao_at_cse.cuhk.edu.hk
Department of Computer Science Engineering
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Outline

• Review of P, NP, NP-C
• 2 problems
• Double-SAT
• Dominating set http//en.wikipedia.org/wiki/Domina
  ting_set_problem

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Relations
hard
NP-C
Is there any problem even harder than NP-C?
NP
Yes! e.g. I-go
P
easy
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Polynomial Time Reduction
How to show that a problem R is not easier than a
problem Q?
Informally, if R can be solved efficiently, we
can solve Q efficiently.

• Formally, we say Q polynomially reduces to R if
• Given an instance q of problem Q
• There is a polynomial time transformation to an
  instance f(q) of R
• q is a yes instance if and only if f(q) is a
  yes instance

Then, if R is polynomial time solvable, then Q is
polynomial time solvable.
If Q is not polynomial time solvable, then R is
not polynomial time solvable.
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Methodology

• To show L is in NP, you can either (i) show that
  solutions for L can be verified in
  polynomial-time, or (ii) describe a
  nondeterministic polynomial-time TM for L.
• To show L is NP-complete, you have to design a
  polynomial-time reduction from some problem we
  know to be NP-complete

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• The direction of the reduction is very important
• Saying A is easier than B and B is easier than
  A mean different things
• What we have? We know SAT, Vertex Cover problems
  are NP-Complete!

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Double-SAT

• Definition
• Double-SAT ltfgt f is a Boolean formula with
  at least two satisfying assignments
• Show that Double-SAT is NP-Complete.
• (1) First, it is easy to see that Double-SAT ?
  NP.
• non-deterministically guess 2 assignments for f
  and verify whether both satisfy f.
• (2) Then we show Double-SAT is not easier than
  SAT.
• Reduction from SAT to Double-SAT

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Double-SAT

• Reduction
• On input f(x1, . . . , xn)
• 1. Introduce a new variable y.
• 2. Output formula
• f(x1, . . . , xn, y) f(x1, . .
  . , xn) ? ( y ? y ).

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Dominating set

• Definition input G(V,E), K
• Let G(V,E) be an undirected graph. A dominating
  set D is a set of vertices in G such that every
  vertex of G is either in D or is adjacent to at
  least one vertex from D. The problem is to
  determine whether there is a dominating set of
  size K for G.

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Dominating set - example

• yellow vertices is an example of a dominating
  set of size 2.

e
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Dominating set

• Show that Dominating set is NP-Complete.
• (1) First, it is easy to see that Dominating set
  ? NP.
• Given a vertex set D of size K, we check whether
  (V-D) are adjacent to D.
• (2) Then we show Dominating set is not easier
  than Vertex cover.
• Reduction from Vertex cover to Dominating set

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Dominating set

• Reduction
• (1) Graph transformation - Construct a new graph
  G' by adding new vertices and edges to the graph
  G as follows For each edge (v, w) of G, add a
  vertex vw and the edges (v, vw) and (w, vw) to G'
  . Furthermore, remove all vertices with no
  incident edges such vertices would always have
  to go in a dominating set but are not needed in a
  vertex cover of G.

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Dominating set graph transformation
vw
v
w
w
v
vz
wu
vu
z
u
z
u
G
zu
G'
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Dominating set

• Reduction
• (1) Graph transformation
• (2) a dominating set of size K in G ?? a vertex
  cover of size K in G

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• Thank you!