Computational Complexity

Shirley Moorehttp://www.cs.utep.edu/svmoore

CS4390/5390 Fall 2013August 27, 2013

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

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)

Preliminaries

• Course website: http://svmoore.pbworks.com/

• Syllabus• Pre-course survey

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)

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)

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?

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