Computational Complexity: P, NP, and NP-Complete, Slides of Algorithms and Programming

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P, NP, and NP-Complete

Dr. Ying Lu

ylu@cse.unl.edu

RAIK 283

Data Structures & Algorithms

RAIK 283

Data Structures & Algorithms

Giving credit where credit is due:

» Most of the lecture notes are based on slides

created by Dr. Cusack and Dr. Leubke.

» I have modified them and added new slides

Polynomial-Time Algorithms



Are some problems solvable in polynomial time?

» (^) Of course: many algorithms we’ve studied provide polynomial-time solutions to some problems 

Are all problems solvable in polynomial time?

» (^) No: Turing’s “Halting Problem” is not solvable by any computer, no matter how much time is given 

Most problems that do not yield polynomial-time

algorithms are either optimization or decision

problems.

Optimization/Decision

Problems



Optimization Problems

» (^) An optimization problem is one which asks, “What is the optimal solution to problem X?” » (^) Examples:  (^) 0-1 Knapsack  (^) Fractional Knapsack  (^) Minimum Spanning Tree

 Decision Problems

» (^) An decision problem is one with yes/no answer » (^) Examples:  (^) Does a graph G have a MST of weight  W?

The Class P

P : the class of decision problems that have polynomial-time deterministic algorithms. » (^) That is, they are solvable in O ( p ( n )), where p ( n ) is a polynomial on n » (^) A deterministic algorithm is (essentially) one that always computes the correct answer Why polynomial? » (^) if not, very inefficient » (^) nice closure properties  (^) the sum and composition of two polynomials are always polynomials too

Sample Problems in P



Fractional Knapsack



MST



Sorting



Others?

Sample Problems in NP



Fractional Knapsack



MST



Sorting



Others?

» (^) Hamiltonian Cycle (Traveling Salesman) » (^) Graph Coloring » (^) Satisfiability (SAT)  (^) the problem of deciding whether a given Boolean formula is satisfiable

The Satisfiability (SAT)

Problem



Satisfiability (SAT):

» (^) Given a Boolean expression on n variables, can we assign values such that the expression is TRUE? » (^) Ex: (( x 1  x 2 )  (( x 1  x 3 )  x 4 ))  x 2 » (^) Seems simple enough, but no known deterministic polynomial time algorithm exists » (^) Easy to verify in polynomial time!

Review: P And NP Summary

 P = set of problems that can be solved in polynomial time

» (^) Examples: Fractional Knapsack, …

 NP = set of problems for which a solution can be verified

in polynomial time

» (^) Examples: Fractional Knapsack,…, Hamiltonian Cycle, CNF SAT, 3-CNF SAT

 Clearly P  NP

 Open question: Does P = NP?

» (^) Most suspect not » (^) An August 2010 claim of proof that P ≠ NP, by Vinay Deolalikar, researcher at HP Labs, Palo Alto, has flaws

NP -complete problems



A decision problem D is NP -complete iff

1. D  NP

2. every problem in NP is polynomial-time reducible to D 

Cook’s theorem (1971): CNF-sat is NP -complete

NP-Hard and NP-Complete



If R is polynomial-time reducible to Q, we denote

this R p Q



Definition of NP-Hard and NP-Complete:

» (^) If all problems R  NP are polynomial-time reducible to Q, then Q is NP-Hard » (^) We say Q is NP-Complete if Q is NP-Hard and Q  NP

 If R 

p Q and R is NP-Hard, Q is also NP-Hard

(why?)

Practical Implication



Given a problem that is known to be NP-Complete

» (^) Try to solve it by designing a polynomial-time algorithm?  (^) Prove P=NP » (^) Alleviate the intractability of such problems  (^) To make some large instances of the problem solvable (like solving some instances of Knapsack problem in polynomial time)  (^) To find good approximations (Chap 12)