A Personal view of P versus NP
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A PERSONAL VIEW OF P VERSUS NP
Lance FortnowGeorgia Institute of Technology
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Step 1: Post Elusive Proof. Step 2: Watch Fireworks. By John Markoff
… Vinay Deolalikar, a mathematician and electrical engineer at Hewlett-Packard, posted a proposed proof of what is known as the “P versus NP” problem on a Web site, and quietly notified a number of the key researchers.
Email: August 6, 2010From: Deolalikar, VinayTo: 22 people Dear Fellow Researchers,
I am pleased to announce a proof that P is not equal to NP, which is attached in 10pt and 12pt fonts…
NEW YORK TIMES AUGUST 16, 2010
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$1 Million Award for solving any of these problems. Birch and Swinnerton-Dyer
Conjecture Hodge Conjecture Navier-Stokes Equations P vs NP Poincaré Conjecture Riemann Hypothesis Yang-Mills Theory
CLAY MATH MILLENNIUM PRIZES
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FRIENDS AND ENEMIES
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FRIENDS AND ENEMIES OF FRENEMY
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DATING SERVICE
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DATING SERVICE
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DATING SERVICE
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DATING SERVICE
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DATING SERVICE
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DATING SERVICE
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DATING SERVICE
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DATING SERVICE
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DATING SERVICE
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We can efficiently find a matching even among millions of men and women avoiding having to search all the possibilities.
EFFICIENT ALGORITHMS
P
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CLIQUE
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CLIQUE
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CLIQUE
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CLIQUE: HARD TO FIND
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CLIQUE: EASY TO VERIFY
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Given a solution to a clique problem we can check it quickly
EFFICIENTLY VERIFIABLE
NP
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EASY TO SOLVE
EASY TO VERIFY
P AND NP
NPP
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EVERY PROBLEM WE CAN VERIFY EFFICIENTLY WE CAN SOLVE EFFICIENTLY
P = NP
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THERE ARE PROBLEMS WE CAN VERIFY QUICKLY THAT WE CAN’T SOLVE QUICKLY
P ≠ NP
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CAN WE SOLVE EVERY PROBLEM QUICKLY IF THE SOLUTIONS ARE EASILY VERIFIABLE?
P = NP?
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WRITING ABOUT P AND NP
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THE P VERSUS NP PROBLEM
Two views of the problem Mathematical
) = ) ? World View
Can we “efficiently” solve all problems where we can “efficiently” check the solutions?
How does the world change if P = NP? How do we deal with hard problems if P ≠ NP?
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MATHEMATICAL VIEW OF P VS NP
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TURING MACHINE
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FORMALIZING THE TURING MACHINE
Transition Function
Tape AlphabetBlank Symbol
Input Alphabet
State SpaceStart State
Accept State
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TRANSITIONS
Transition function (state, symbol) →(state, symbol, direction)
Nondeterministic Can map to multiple possibilities
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DEFINING P AND NP
DTIME(t(n)) is the set of languages accepted by deterministic Turing machines in time t(n)
NTIME(t(n)) is the set of languages accepted by nondeterministic Turing machines in time t(n)
P = ) NP = )
Does P = NP?
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MATHEMATICALLY ROBUST
Instead of Turing machine Multiple tapes Random access λ – calculus C++ LaTeX
Probabilistic and Quantum computers might not define the same class
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REDUCTIONS
A B
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NP-COMPLETE
Hardest problems in NP Cook-Levin 1971
Boolean Formula Satisfiability
u v w u w x v w x
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NP-COMPLETE
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1935: Turing’s Machine 1962: Hartmanis-Stearns: Computation time
depends on size of problem 1966: Edmonds, Cobham: Models of efficient
computation 1971: Steve Cook defines first NP-complete
problem 1972: Richard Karp shows 22 common problems
NP-complete 1971: Leonid Levin similar work in Russia 1979: Garey and Johnson publish list of 100’s of
NP-complete problems Now thousands of NP-complete problems over
many disciplines
VERY SHORT HISTORY
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OUTSIDE WORLD OF P VERSUS NP
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WHAT HAPPENS IF P = NP?
WECURE
CANCER
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CURING CANCER
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William of Ockham, English Franciscan Friar Occam’s Razor (14th Century) Entia non sunt
multiplicanda praeter necessitatem
OCCAM’S RAZOR
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William of Ockham English Franciscan Friar Occam’s Razor (14th Century) Entities must not be
multiplied beyond necessity
The simplest explanation is usually the best.
If P = NP we can find that “simplest explanation”.
OCCAM’S RAZOR
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Rosetta Stone 196 BC Decree in three
languages• Greek• Deomotic• Hieroglyphic In 1822, Jean-François
Champollion found a simple grammar.
TRANSLATION
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MACHINE LEARNING
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IF P = NP
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IF P NP: CRYPTOGRAPHY
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IF P NP: ZERO-KNOWLEDGE PROOFS
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DEALING WITH HARDNESS
How do you deal with NP-completeness?
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Brute Force Heuristics Small Parameters Approximation Solve a Different Problem Give Up
DEALING WITH HARDNESS
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HOW DO WE PROVE P ≠ NP?
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WHAT DOESN’T WORK?
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DIAGONALIZATION
1 2 3 4 5 6
S1 In Out In Out In In
S2 Out In Out Out In Out
S3 Out Out Out Out Out Out
S4 In Out In Out In Out
S5 In In In In In In
S6 Out In Out Out Out In
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DIAGONALIZATION
NP doesn’t have enough power to simulate P
Relativized world where P = NP. Can get weaker time/space results:
No algorithm for satisfiability that uses logarithmic space and n1.8 time.
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CIRCUIT COMPLEXITY
Measure complexity by size of circuit. Different circuits for each input length. Efficient computation essentially equivalent
to small circuits.
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CIRCUITS
Idea: Show no single gate changes things much so needs lots of gates for NP-complete problems
Works for circuits of limited depth or negations.
“Natural Proofs” give some limitations on this technique.
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PROOF COMPLEXITY
( x AND y ) OR (NOT x) OR (Not y)
If P = NP (or even NP = co-NP) then every tautology has a short proof.
Try to show tautologies only have long proofs.
Works only for limited proof systems like resolution.
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THE FUTURE OF P V NP
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THE GOLDEN TICKET
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