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NP-Completeness

November 28, 2003

Young Eun Kim and Gene Moo Lee

Department of Computer Science & EngineeringKorea University

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Contents• Introduction and Motivation• Background Knowledge• Definition of NP-Completeness• Examples of NP-Complete Problems• Hierarchy of Problems• How to Prove NP-Completeness• How to Cope with NP-Complete Problems• Conclusion

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Introduction (1/2)

• Some Algorithms we’ve seen in this class– Sorting – O(N log N)– Searching – O(log N)– Shortest Path Finding – O(N2)

• However, some problems only have– Exponential Time Algorithm O(2N) – So What?

Why? What? How?

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Introduction (2/2)

N 10 20 30 40 50 60O(N) .00001

second.00002second

.00003second

.00004second

.00005second

.00006second

O(N2) .0001second

.0004second

.0009second

.0016second

.0025second

.0036second

O(N3) .001second

.008second

.027second

.064second

.125second

.216second

O(N5) 1second

3.2seconds

24.3seconds

1.7minutes

5.2minutes

13.0minutes

O(2N) .001second

1.0second

17.9minutes

12.7days

35.7years

366centuries

O(3N) .059second

58minutes

6.5years

3855centuries

2*108

centuries1013

centuries

Why? What? How?

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Motivation• Traveling Salesman Problem (n = 1000)

• Compute 1000!

– Even Electron in the Universe is a Super Computer, – And they work for the Estimated Life of the Universe,– WE CANNOT SOLVE THIS PROBLEM!!!– This kind of problems are NP-Complete.

Why? What? How?

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Background KnowledgeTo understand NP-Completeness, need to

know these concepts1. Decision and Optimization Problems2. Turing Machine and class P3. Nondeterminism and class NP4. Polynomial Time Reduction

(Problem Transformation)

Why? What? How?

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Decision and Optimization Problems• What is the Shortest Path from A to B?

– This is an Optimization Problem.• Is there a Path from A to B consisting of at

most K edges?– This is the related Decision Problem.

We consider only Decision Problems!

Why? What? How?

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Turing Machine and Class P (1/3)

Church – Turing Thesis

“Computer ≡ Turing Machine”

Alan Turing(1912-1954)

(www.time.com)

Why? What? How?

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Turing Machine and Class P (2/3)

A Turing machine

is a 7-tuple (Q, ∑, Γ, δ, q0, qaccept, qreject).

Why? What? How?

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Turing Machine and Class P (3/3)

P : the class of problems that are decidable in polynomial time on a Turing machine

Sorting, Shortest Path are in P!

Why? What? How?

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Nondeterminism and Class NP (1/2)• A Nondeterministic Turing machine is a Turing machine with the transition function has the form

δ : Q * Γ P(Q * Γ * {L, R}). • NTM guess to choose the answer nondeterministically

Why? What? How?

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Nondeterminism and Class NP (2/2)• NP : the class of problems that are decidable in

polynomial time on a nondeterministic Turing machine

• Solutions of problems in NP can be checked (verified) in polynomial time.

• If a Hamiltonian path were discovered somehow, we could easily check if the path is Hamiltonian. – HAMPATH is in NP! Also Sorting is in NP!

Why? What? How?

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Class P and NP

• P = the class of problems where membership can be decided quickly.

• NP = the class of problems where membership can be verified quickly.

Why? What? How?

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Polynomial Time Reduction

Traveling Salesman Problem5-CliqueMap Coloring

Traveling Salesman Problem 5-Clique Map Coloring

Why? What? How?

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Definition of NP-Completeness

A problem B is NP-complete if it satisfies two conditions:

1. B is in NP, and2. Every problem A in NP is polynomial

time reducible to B. (NP-Hard)

Why? What? How?

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Meaning of NP-Completeness

• NP: Nondeterministic Polynomial• Complete:

– If one of the problems in NPC have an efficient algorithm, then all the problems in NP have efficient algorithms.

Why? What? How?

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Examples of NP-Completeness

• Satisfiability (SAT)• Traveling Salesman Problem (TSP)• Longest Path (vs. Shortest Path is in P)• Real-Time Scheduling• Hamiltonian Path (vs. Euler Path is in P)

Why? What? How?

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Where are we?

Background KnowledgeDefinition of NP-CompletenessExamples of NP-Complete Problems• Hierarchy of Problems• How to Prove NP-Completeness• How to Cope with NP-Completeness

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Hierarchy of Problems (1/2)

P NP PSPACE = NPSPACE EXPTIME ⊆ ⊆ ⊆

Conjectured Relationships

EXPTIME

NPSPACE

NP PUNDECIDABLE

Why? What? How?

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Hierarchy of Problems (2/2)

NP P

Which one is correct?

An efficient algorithm on a deterministic machine does not exist.

An efficient algorithm on a deterministic machine is not found yet.

P = NP

Why? What? How?

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How to prove NP-Completeness(1/3)

• If B is NP-complete and B C for C in NP, ∝then C is NP-complete.

B C

NP-complete NP-complete

Then, we need at least one NP-complete problem!

Why? What? How?

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How to prove NP-Completeness(2/3)

Cook’s Theorem“Satisfiability Problem (SAT) is

NP-complete.” - the first NP-complete problem!

Stephen Cook (www.cs.toronto.edu)

Why? What? How?

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I’m also NP Complete!

How to prove NP-Completeness(3/3)

any NP problem

can be reduced to...

SAT

new NP problem

can be reduced to...Proved by

CookNew Problem is “no easier” than SAT

Why? What? How?

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NP-Complete Problems TreeSAT

3-SAT Graph 3-Color

3-DMExact Cover Planer 3-color

Vertex Cover

Subset-Sum

HAMPATH Clique

Partition Integer Programming

IndependentTSP (Salesman)

Why? What? How?

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How to Cope with NP-Completeness

I. HEURISTIC ALGORITHM To find a solution within a reduced search-space. II. APPROXIMATION ALGORITHM

To find approximately optimal solutions.III. QUANTUM COMPUTING

To use the spins of quantum with the speed of light. (bit 0, 1 spin-up (0), spin-down (1))

Why? What? How?

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Heuristic Algorithm

• In NP-Complete Problems, we have to check exponential possibilities.

• By Heuristic, reduce the search space.

• Example: Practical SAT problem Solvers– zChaff, BerkMin, GRASP, SATO, etc.

Why? What? How?

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Approximation

• Hard to find an exactly correct solution in NP-complete problems

• By Approximation, find a nearly optimal solution.

• Example: finding the smallest vertex covers(we can find a vertex cover never more than twice the size of the smallest one.)

Why? What? How?

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Quantum Computation

• Bit 0 and 1 Spin Up and Spin Down

• Speed of electron Speed of light

Why? What? How?

Digital Comp. Quantum Comp.

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ConclusionWhen a hard problem is given,we can prove that a problem is NP-complete,

just by finding a polynomial time reduction. After proving,we can solve the problem in these ways:Heuristic Algorithm, Approximation

Algorithm and Quantum computing.

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Thank You for Listening.

Any Question?