Algorithms
NP Completeness Continued
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NP Completeness - Introduction to Algorithms - Lecture Slides, Slides of Computer Science
These are the Lecture Slides of Introduction to Algorithms which includes Expensive Operations, Sort Edges, Running Time, Upshot, Union, Makeset, Disjoint Set, Disjoint Set Union, Naïve Implementation etc. Key important points are: Np Completeness, Tractibility, Undecidable, Computer, Halting Problem, Grow Large, Reasonable Time, Polynomial Time, Ordinary Computer, Unlimited Memory
Typology: Slides
2012/2013
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Algorithms
NP Completeness Continued
Homework 5
● Extension: due midnight Monday 22 April
Review: P
● Some problems are provably decidable in polynomial time on an ordinary computer ■ We say such problems belong to the set P ■ Technically, a computer with unlimited memory ■ How do we typically prove a problem ∈ P?
Review: NP
● Some problems are provably decidable in polynomial time on a nondeterministic computer ■ We say such problems belong to the set NP ■ Can think of a nondeterministic computer as a parallel machine that can freely spawn an infinite number of processes ■ How do we typically prove a problem ∈ NP?
● Is P ⊆ NP? Why or why not?
Review: NP-Complete Problems
● The NP-Complete problems are an interesting class of problems whose status is unknown ■ No polynomial-time algorithm has been discovered for an NP-Complete problem ■ No suprapolynomial lower bound has been proved for any NP-Complete problem, either
● Intuitively and informally, what does it mean for a problem to be NP-Complete?
Review: Reduction
● A problem P can be reduced to another problem Q if any instance of P can be rephrased to an instance of Q, the solution to which provides a solution to the instance of P ■ This rephrasing is called a transformation
● Intuitively: If P reduces in polynomial time to Q, P is “no harder to solve” than Q
NP-Hard and NP-Complete
● If P is polynomial-time reducible to Q, we denote this P ≤p Q
● Definition of NP-Hard and NP-Complete:
■ If all problems R ∈ NP are reducible to P, then P is NP-Hard ■ We say P is NP-Complete if P is NP-Hard and P ∈ NP ■ Note: I got this slightly wrong Friday
● If P ≤p Q and P is NP-Complete, Q is also NP- Complete Docsity.com
Why Prove NP-Completeness?
● Though nobody has proven that P != NP , if you prove a problem NP-Complete, most people accept that it is probably intractable
● Therefore it can be important to prove that a problem is NP-Complete ■ Don’t need to come up with an efficient algorithm ■ Can instead work on approximation algorithms