Approximation
Algorithms for NP-
Hard Problems
D. WINSTON PAUL
AP/IT
SKCET
Pre-requisites
P
NP
–
NP-Complete
–
NP-Hard
3
NP-completeness
Do your best then.
!
4
Coping With NP-Hardness
Coping With NP-Hardness
Brute-force algorithms.
–
Develop clever enumeration strategies.
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Guaranteed to find optimal solution.
–
No guarantees on running time.
Heuristics.
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Develop intuitive (innovative) algorithms.
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Guaranteed to run in polynomial time.
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No guarantees on quality of solution.
Approximation algorithms.
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Guaranteed to run in polynomial time.
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Guaranteed to find "high quality" solution, say within 1% of optimum.
–
Obstacle:
need to prove a solution’s value is close to optimum,
without even knowing what optimum value is!
5
Motivation
By now we’ve seen many
NP-Complete
problems.
We conjecture none of them has polynomial time algorithm.
6MotivationIs this a dead-end? Should we give up altogether??
Complexity
©D Moshkovitz
7
Motivation
Or maybe we can settle for good approximation
algorithms?
Approximation Algorithms
IS CLOSE
ENOUGH GOOD
ENOUGH?
Spring 2003
Approximation Algorithmes
10
Performance ratios
We are going to find a Near-Optimal solution for a given
problem.
We assume two hypothesis :
Each potential solution has a positive cost.
