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# NP-complete examples - PowerPoint PPT Presentation

Fall 2009. NP-complete examples. Xiao Linfu. CSC3130 Tutorial 11. [email protected] Department of Computer Science & Engineering. Outline. Review of P, NP, NP-C 2 problem s Double-SAT Dominating set http://en.wikipedia.org/wiki/Dominating_set_problem. Relations. hard. NP-C.

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### NP-complete examples

Xiao Linfu

CSC3130 Tutorial 11

Department of Computer Science & Engineering

• Review of P, NP, NP-C

• 2 problems

• Double-SAT

• Dominating sethttp://en.wikipedia.org/wiki/Dominating_set_problem

hard

NP-C

Is there any problem even harder than NP-C?

NP

Yes! e.g. I-go

P

easy

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 iff(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.

• 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

• 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!

• Definition:

• Double-SAT = {<φ> |φ 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 φand verify whetherboth satisfy φ.

• (2) Then we show Double-SAT is not easier than SAT.

• Reduction from SAT to Double-SAT

• Reduction:

• On input φ(x1, . . . , xn):

• 1. Introduce a new variable y.

• 2. Output formula

φ’(x1, . . . , xn, y) = φ(x1, . . . , xn) ∧(y∨ y ).

• 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.

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

e

• 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

• 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.

vw

v

w

v

w

vz

wu

vu

z

u

z

u

G

zu

G'

• Reduction

• (1) Graph transformation

• (2) a dominating set of size K in G’  a vertex cover of size K in G