An Introduction to Computational Complexity

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An Introduction to Computational Complexity

• Lance Fortnow
• University of Chicago

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The Round Table Problem
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Sir Percivale
Sir Lancelot
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Sir Gareth
Sir Galahad
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Sir Pericvale
Sir Bevidere
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The Round Table Problem
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Map Coloring
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5 Colors
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All Maps are Four Colorable
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Can we use only 3 colors?
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Sudoku
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Sudoku
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All The Same Problem

• Solve one of these problems, you solve them all.

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And Thousands More

• Minesweeper
• Traveling Salesperson
• Cliques
• Integer Programming
• Quadratic Programming
• Maximum Cut
• Longest Path

• Bin Packing
• Job Scheduling
• Filling a Knapsack
• Crossword Puzzle Construction
• Code Generation
• Sequencing DNA
• Finding Arbitrage

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Can We Solve These Problems?

• P NP?

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The P versus NP Question

• P Efficiently Computable Solutions
• NP Efficiently Verifiable Solutions

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P Efficiently Computable

• Finding the shortest path from point A to point B

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NP Efficiently Verifiable
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NP Efficiently Verifiable
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NP-Complete
NPC
NP
P
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If P ? NP
NPC
NP
P
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If P NP
NPC
NP
P
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P versus NP

• General Consensus P ? NP
• Major Open Problem Formally prove P ? NP
• First to do so wins 1,000,000
• One of the seven Millennium Problems of the
  Clay Mathematics Institute.
• Many approaches, all have failed miserably.

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P versus NP A Short History

• 1936 Alan Turing tries to understand how a
  mathematician thinks and developed a theoretical
  model of computation we now call the Turing
  Machine.

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P versus NP A Short History

• 1956 The logician Kurt Gödel, in a letter to
  John van Neumann, asked a question mathematically
  equivalent to P NP?
• The letter was never followed up and soon
  forgotten.

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P versus NP A Short History

• 1962 Juris Hartmanis and Richard Stearns develop
  the notion of Computational Complexity by
  suggesting we look at the time a program takes as
  a function of the size of the input.
• They show that given more time, a computer can do
  solve more problems.

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P versus NP A Short History

• 1965 Independently Alan Cobham and Jack Edmonds
  suggest that one can characterize efficient
  computation by having the running time run in
  some fixed polynomial of the input length.
• This class of problems is later given the name P.

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P versus NP A Short History

• 1972 Steve Cook gave the first NP-complete
  problem, Boolean Formula Satisfiability, and
  formally defined the P versus NP problem.

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P versus NP A Short History

• 1973 Richard Karp showed that several natural
  problems, such as Clique and Traveling
  Salesperson are NP-complete.
• The rest is history.

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Suppose P NP

• What a beautiful world

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We Can Efficiently Solve

• Minesweeper
• Traveling Salesperson
• Cliques
• Integer Programming
• Quadratic Programming
• Maximum Cut
• Longest Path

• Bin Packing
• Job Scheduling
• Filling a Knapsack
• Crossword Puzzle Construction
• Code Generation
• Sequencing DNA
• Finding Arbitrage

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Occams Razor

• William of Ockham
• Entities should not be multiplied beyond
  necessity.
• The shortest explanation of the data is the best
  one.

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Translation

• In the beginning God created the heavens and the
  earth. Now the earth was formless and empty,
  darkness was over the surface of the deep, and
  the Spirit of God was hovering over the waters.
  And God said, "Let there be light," and there was
  light. God saw that the light was good, and He
  separated the light from the darkness. God called
  the light "day," and the darkness he called
  "night." And there was evening, and there was
  morningthe first day.

• Im Anfang schuf Gott die Himmel und die Erde. Und
  die Erde war wüst und leer, und Finsternis war
  über der Tiefe und der Geist Gottes schwebte
  über den Wassern. Und Gott sprach Es werde
  Licht! Und es wurde Licht. Und Gott sah das
  Licht, daß es gut war und Gott schied das Licht
  von der Finsternis. Und Gott nannte das Licht
  Tag, und die Finsternis nannte er Nacht. Und es
  wurde Abend, und es wurde Morgen ein Tag.

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Translation

• In the beginning God created the heavens and the
  earth. Now the earth was formless and empty,
  darkness was over the surface of the deep, and
  the Spirit of God was hovering over the waters.
  And God said, "Let there be light," and there was
  light. God saw that the light was good, and He
  separated the light from the darkness. God called
  the light "day," and the darkness he called
  "night." And there was evening, and there was
  morningthe first day.

• Im Anfang schuf Gott die Himmel und die Erde. Und
  die Erde war wüst und leer, und Finsternis war
  über der Tiefe und der Geist Gottes schwebte
  über den Wassern. Und Gott sprach Es werde
  Licht! Und es wurde Licht. Und Gott sah das
  Licht, daß es gut war und Gott schied das Licht
  von der Finsternis. Und Gott nannte das Licht
  Tag, und die Finsternis nannte er Nacht. Und es
  wurde Abend, und es wurde Morgen ein Tag.

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If P NP

• Can efficiently find the smallest program to
  translate documents.
• Recognizing Faces
• Speech to Text, Pictures to Text
• Recognizing Music

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No Such Luck

• Life cant be so good so we all believeP ? NP.
• Once we show a problem is NP-Complete we should
  stop looking for an efficient algorithm.

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What to do about hard problems?

• Solve it anyway.
• Computers are very fast these days.
• Heuristics.
• Your specific instance might not be that hard.
• Approximation.
• Cant get best algorithm but perhaps one can get
  a good enough solution.

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Solve It Anyway

• There are about 2x1075 possible numbers to try.
• A simple backtracking algorithm can solve the
  problem on a PC in a few seconds.

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Heuristics

• Most puzzles are designed to be solved by humans.

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Approximation
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Clique Problem

• Find a large group of people all of whom are
  friends with each other.
• Coming up with a clique anywhere close to the
  largest clique is as hard as solving NP-complete
  problems.

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Good News with Hard Problems

• Pseudorandom Generators
• Hard functions can help us create fake random
  coins that are as good as real random coins.
• Public-Key Cryptography

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Public-Key Cryptography
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Impossible if P NP
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Possible if Factoring is Hard
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NP as Proof System
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NP as Proof System
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Zero-Knowledge Proof System

• Can convince someone that a solution exists
  without revealing any details of the solution.
• Requires cryptography and verification process is
  randomized.

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Interactive Proofs

• Can convince someone there is no solution.
• Verification process is randomized.

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5
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Proof Checking
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Probabilistically Checkable Proof

• There is a proof that a solution exists where we
  need to only see three randomly chosen bits of
  the proof.
• This is the tool that shows that Clique is hard
  to approximate.

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Quantum Computing
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Quantum Bits
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Entanglement
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Interference

• Use entanglement to search many states.
• Use interference to eliminate the ones that dont
  solve your problem.

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Shors Algorithm

• Factor quickly with a quantum computer.

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Complexity of Factoring
NPC
NP
P
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Complexity of Quantum
NPC
NP
Quantum
P
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Grovers Algorithm

• Search a trillion pieces of hay to find the
  needle by making only a million quantum hay
  searches.
• Can be used to give quadratic improvement to
  solving NP problems.

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Quantum Computers in Our Future?

• A very very long way from building quantum
  computers that can outperform current machines in
  any task.
• Even if we build quantum computers, factoring is
  only good for breaking some codes. Quantum
  computers could end up being a very specialized
  tool.

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The Future of Complexity

• Solve the P versus NP problem.
• Understanding complexity issues in the
  computational processes outside of traditional
  digital computers.
• Biological Systems (Protein Folding)
• Weather
• Social Interactions
• Financial Markets