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NP Complete

Joe Meehean

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Complexities we’ve seen

Log: binary search in sorted array

Linear: traverse a tree

Log-Linear: insert into a heap

Quadratic (N2): your sort from P1 Exponential (NN): brute force

map coloring

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Polynomial Time Upper bound is Nk for N inputs

• e.g., O(logN), O(N), O(NlogN), O(N2), … Problems in class P

• have known polynomial time solutions

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Non-deterministic vs. Deterministic

Slipping into computational theory Deterministic programs

• like the ones we write in C++• must search or construct the correct answer• if there are many possible solutions must

check them all• OR carefully construct the answer piece-

wise

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Non-deterministic vs. Deterministic

Non-deterministic programs• currently exist only in theory• program guided by all knowing oracle• oracle can create a possible solution to the

problem for free• solution is often the correct one

if a solution exists

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Non-deterministic Polynomial Time

Think about reducing problem to yes/no

Given a potential solution to problem• is it possible to determine whether it is

correct in polynomial time Problems like this are in the NP class

• if in NP, but not P then there is no known polynomial time solution for these problems

• but we can check answers in polynomial time

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Non-deterministic Polynomial Time

e.g., graph coloring• given a potential coloring of vertices• for each vertex compare its colors to

its neighbors • O(E + V) = polynomial time• only need one correct k coloring to prove

graph can be colored with k colors

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Problem Complexity

NP

P

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NP Complete Problems Hardest problems in NP Any problem in NP can be polynomially

reduced to all NP complete problems• NP problem A can be translated into NP

complete problem B in polynomial time• e.g. calculator

numbers entered in decimal converted to binary and solved in binary converted back to decimal

Min color graph coloring is NP complete

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Problem Complexity

NP

P NP Complete

maybe

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NP Complete If we can solve an NP complete

problem A in polynomial time• we can use it as a method (sub routine) to

solve other NP problem B• just translate the B into A in polynomial

time• solve A• translate solution back into B• polynomial time + polynomial time =

polynomial time

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NP Complete If we can solve an NP complete

problem A in polynomial time• we can solve ALL NP problems in

polynomial time• P = NP

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Problem Complexity

NP

P

NP Complete

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NP Complete P = NP? No one has solved an NP complete

problem in polynomial time …Yet

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Undecidable problems Problems that are impossible to

solve using a Turing machine-based computer• impossible to solve with modern

computers E.g., halting problem

• does a program have an infinite loop?• proof is simple to explain• proof is hard to understand

involves recursion and impossible outcomes

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Questions?