NP-Complete Problems - Automata and Complexity Theory - Lecture Slides, Slides of Theory of Automata

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More NP-Complete Problems

NP-Hard Problems Tautology Problem

Node Cover

Knapsack

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Next Steps

We can now reduce 3SAT to a large number of problems, either directly orindirectly.

Each reduction must be polytime.

Usually we focus on length of the output from the transducer, becausethe construction is easy.

But key issue: must be polytime.

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Example: NP-Hard Problem

The

Tautology Problem

is: given a

Boolean formula, is it satisfied by alltruth assignments?



Example: x + -x + yz

Not obviously in

NP

, but it’s

complement is.



Guess a truth assignment; accept if thatassignment doesn’t satisfy the formula.

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Key Point Regarding Tautology 

An NTM can guess a truth assignment and decide whether formula F issatisfied by that assignment inpolytime.

But if the NTM accepts when it guesses a satisfying assignment, it will accept Fwhenever F is in SAT, not Tautology.

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Tautology is NP-Hard

While we can’t prove Tautology is in NP

, we can prove it is NP-hard.

Suppose we had a polytime algorithm for Tautology.

Take any Boolean formula F and convert it to -(F).



Obviously linear time.

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Tautology is NP-Hard – (2)

F is satisfiable if and only -(F) is not a tautology.

Use the hypothetical polytime algorithm for Tautology to test if -(F) is atautology.

Say “yes, F is in SAT” if -(F) is not a tautology and say “no” otherwise.

Then SAT would be in

P

, and

P

NP

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History – (2)

Cook used “if problem X is in

P

, then

P

NP

” as the definition of “X is NP-

hard.”



Today called

Cook completeness.

In 1972, Richard Karp wrote a paper showing many of the key problems inOperations Research to be NP-complete.

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History – (3)

Karp’s paper moved “NP-completeness” from a concept about theorem provingto an essential for any study ofalgorithms.

But Karp used the definition of NP- completeness “exists a polytimereduction,” as we have.



Called

Karp completeness.

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Cook Vs. Karp – (2)

But there is one important reason we prefer Karp-completeness.

Suppose I had an algorithm for some NP-complete problem that ran in timeO(n

log n



A function that is bigger than anypolynomial, yet smaller than theexponentials like 2

n

.

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Cook Vs. Karp – (3)

If “NP-complete is Karp-completeness, I can conclude that all of

NP

can be

solved in time O(n

f(n)

), where f(n) is

some function of the form c log

k

n.



Still faster than any exponential, and fasterthan we have a right to expect.

But if I use Cook-completeness, I cannot say anything of this type.

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Example: Node Cover

A

C

E

F

D

B

One possible node coverof size 3: {B, C, E}

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NP-Completeness of Node Cover

Reduction from 3SAT.

For each clause (X+Y+Z) construct a “column” of three nodes, all connectedby

vertical

edges.

Add a

horizontal

edge between nodes

that represent any variable and itsnegation.

Budget = twice the number of clauses.

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Example: Reduction – (2)

A node cover must have at least two nodes from every column, or somevertical edge is not covered.

Since the budget is twice the number of columns, there must be exactly twonodes in the cover from each column.

Satisfying assignment corresponds to the node in each column not selected.

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Example: Reduction – (3)

(x + y + z)(-x + -y + -z)(x + -y +z)(-x + y + -z)Truth assignment: x = y = T; z = F

x y z

-x -y -z

x z -y

-x -z

y

Pick a true node in each column