Analogy (sort of)

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Analogy (sort of)
Lord of The Rings
One ring to rule them all
Computational Complexity
One problem to solve them all
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NP Complexity
The class NP consists of all problems that can be
solved in polynomial time by nondeterministic
algorithms
Nondeterministic algorithms are done in two
phases

• Phase I algorithm that makes a guess (solutions
  must be included among the possible guesses)
• Phase II algorithm that checks if the guess is
  an actual solution or not

3
The Traveling Salesperson Problem (TSP)

• Given the cities and cost to travel between
  cities, obtain a route R such that
• All cities are in R
• Every city is visited only once in R
• R has the minimum cost. That is, for any other
  route R meeting 1-2, cost(R) ? cost(R).

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Example Traveling Salesman Problem (TSP) is in NP
20
G
H
10
16
9
B
6
F
C
8
24
4
A
6
E
D
20
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Proof
We can show that Traveling Salesman Problem (TSP)
is in NP
phaseII(C path, min int ) //input C a guessed
solution //output true iff C is a TSP If C lt
n then return false Visited ? for i 1 to n do
(u,v) Ci if v in Visited then
return false else Visited ?
Visited v return cost(C) min
phaseI(G Graph) //input G a graph with n
nodes //output C a guess for TSP v ?
randomNode(G) C ? () While random(0,1) 1
and C lt n do w ?
PickNeighbourRandomly(v) C ? C (v,w) v
? w return C
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Why do We Care About NP Problems?

• Traveling Salesperson
• Longest path
• Graph coloring

Network Problems
Data Storage

• Minimum Bin Packing

Scheduling

• Minimum Job Scheduling
• Minimum Multiprocessor Scheduling

Mathematical Programming

• Knapsack

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Why do We Care About NP Problems? (II)

• Longest computation
• Shortest computation

Automata Theory (computing)

• Find a Universal Problem Solver

Planning
All of these problems have 4 things in common

• They are important for a group of people

• No knows (for sure) if they are in P or not

• All of them are in NP

• They all are in a special category of NP
  problems NP-Complete problems

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NP-Complete

• A problem npc is NP-complete if
• npc is in NP
• Every other problem prob in NP can be
  transformed in polynomial time into npc. (NP-Hard)

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NP-Complete
npc

1. If we can find one NP-complete problem the can be
   solved in polynomial time then P NP
2. If we can show for one NP-complete problem that
   it cannot be solved in polynomial time then no
   other NP-complete problem can be solved in
   polynomial time (and P ?NP)

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Fine, But How Do one Proof that a Problem is
NP-Complete?

• First problem was hard to proof Conjunctive
  Normal Form (Cook, 1971)
• Every problem q afterwards is easier

• Show that q is in NP
• Find a problem, npc, that is NP-complete and show
  that npc can be transformed in polynomial time
  into q

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Conjunctive Normal Form
A conjunctive normal form (CNF) is a Boolean
expression consisting of one or more disjunctive
formulas connected by an AND symbol (?). A
disjunctive formula is a collection of one or
more (positive and negative) literals connected
by an OR symbol (?).
Example (a) ? ( a ? b ? c ? d) ?
(c ? d) ? (d)
Problem (CNF-satisfaction) Give an algorithm
that receives as input a CNF form and returns
Boolean assignments for each literal in form such
that form is true
Example (above) a ? true, b ?
false, c ? true, d ? false
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Cook Theorem (1971)
The CNF-satisfaction satisfaction is NP-complete