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Computational Complexity

Computational Complexity. Shirley Moore http://www.cs.utep.edu/svmoore CS4390/5390 Fall 2013 August 27, 2013. Learning Outcomes. After completing this lesson, you should be able to Define the complexity classes P and NP Define NP-complete

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Computational Complexity

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  1. Computational Complexity Shirley Moore http://www.cs.utep.edu/svmoore CS4390/5390 Fall 2013 August 27, 2013

  2. Learning Outcomes • After completing this lesson, you should be able to • Define the complexity classes P and NP • Define NP-complete • Discuss implications of computational complexity for cryptography • Apply logical proof techniques in the context of computational complexity

  3. Agenda • Preliminaries (15 min) • Hamiltonian cycle problem (20 min) • P vs. NP (15 min) • In-class exercise (20 min) • Wrapup and homework assignment (10 min)

  4. Preliminaries • Course website: http://svmoore.pbworks.com/ • Syllabus • Pre-course survey

  5. Hamiltonian Cycle Problem Input: A graph G = (V,E) Problem: Does an ordering of the vertices along edges in the graph exist such that each vertex is visited exactly once? In-class exercise (see handout)

  6. P vs. NP Problem • Major unsolved problem in computer science • Problem statement: Can every problem whose solution can be verified in polynomial time by a deterministic algorithm also be solved in polynomial time by a deterministic algorithm? • One of the seven Millennium Prize Problems selected by the Clay Mathematics Institute to give a $1,000,000 prize for the first correct solution (one already solved) • http://www.claymath.org/millennium/ • In-class reading and discussion questions (see handout)

  7. NP-Complete Problems • Definition: A problem L is NP-complete if and only if: • L is in NP • for every problem L’ in NP, there is a polynomial-time reduction from L’ to L • Theorem: Let L be an NP-complete problem. Then P = NP if and only if L is in P. • Proof: In-class exercise • What implication does the above theorem have for attacking the P vs. NP problem?

  8. Homework • Read “The Status of the P Versus NP Problem” by Lance Fortnow and come to class prepared to discuss • http://cacm.acm.org/magazines/2009/9/38904-the-status-of-the-p-versus-np-problem/fulltext • Read “An NP-complete set” and “More NP-Complete Problems” from http://www.cs.uky.edu/~lewis/cs-heuristic/text/contents.html • Attempt to prove that the Hamiltonian Cycle problem is NP-complete

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