Optimization/Decision Problems

Optimization Problems
 An
optimization problem is one which asks, “What is
the optimal solution to problem X?”
 Examples:
 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?
Some problems are intractable:
as they grow large, we are unable to solve them
in reasonable time
•What reasonable time?
Standard working definition: polynomial time
• Polynomial time: O(n2), O(n3), O(1), O(n lg n)
•Not in polynomial time: O(2n), O(nn), O(n!)
Determinism vs. Nondeterminism


Nondeterministic algorithms produce an answer by
a series of “correct guesses”
Deterministic algorithms (like those that a computer
executes) make decisions based on information.
Complexity Class P


Deterministic in nature
Solved by conventional computers in polynomial
time
 O(1)
 O(log
n)
 O(n)
 O(n
log n)
 O(n2)

Constant
Sub-linear
Linear
Nearly Linear
Quadratic
Polynomial upper and lower bounds
Complexity Class NP

1.
2.
Two Properties:
non-deterministic method to generate possible
solutions
deterministic method to verify in polynomial time that
the solution is correct.
Relation of P and NP





P is a subset of NP
“P = NP”?
Language L is in NP, complement of L is in co-NP
co-NP ≠ NP
P ≠ co-NP
TSP
2
1
3
4
2
1
1
5
4
1
2
2
2
1
i = 23
3
4
2
2
1
1

For each two cities, an integer cost is given to travel from one
of the two cities to the other. The salesperson wants to make a
minimum cost circuit visiting each city exactly once.
NP-Complete
“NP-Complete” comes from:
 Nondeterministic Polynomial
 Complete - “Solve one, Solve them all”
There are more NP-Complete problems than provably
intractable problems.