Algorithm Design and Analysis

Algorithm Analysis and Design NP-Completeness
Pham Quang Dung

Hanoi, 2012

Pham Quang Dung ()

Algorithm Analysis and Design NP-Completeness

Hanoi, 2012

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Outline

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Easy problems - class P Decision problems vs. Optimization problems Class NP Reductions NP-complete class

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Pham Quang Dung ()

Algorithm Analysis and Design NP-Completeness

Hanoi, 2012

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Class P: Problems that are solvable by polynomial-time algorithms (O(nk ) where n is the size of the input to the problem and k is some constant) Examples:
Sorting Shortest path problems Minimum Spanning Tree problems Maximum Flow problems ...

Pham Quang Dung ()

Algorithm Analysis and Design NP-Completeness

Hanoi, 2012

3 / 31

Outline

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Easy problems - class P Decision problems vs. Optimization problems Class NP Reductions NP-complete class

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Pham Quang Dung ()

Algorithm Analysis and Design NP-Completeness

Hanoi, 2012

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Decision problems

Answer is “yes” or “no”
Is n a prime number? Does an undirected graph contain a hamilton cycle? Satisfiability: Given a boolean formula. Does there exist some assignment of the values 0 and 1 to its variables such that the value of the formula is 1? Subset sum problem: Given a set of integers S = {x1 , . . . , xn } and an integer T . Is there a subset of S whose sum is equal to T ?

Pham Quang Dung ()

Algorithm Analysis and Design NP-Completeness

Hanoi, 2012

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Optimization problems

Each feasible solution is associated with a value Objective: Find a feasible solution with the best value
Shortest path problem Minimum Spanning Tree problem Traveling Salemans Problem (TSP)

Pham Quang Dung ()

Algorithm Analysis and Design NP-Completeness

Hanoi, 2012

6 / 31

Outline

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Easy problems - class P Decision problems vs. Optimization problems Class NP Reductions NP-complete class

2

3

4

5

Pham Quang Dung ()