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Lance Fortnow Georgia Institute of Technology. A Personal view of P versus NP. Step 1: Post Elusive Proof. Step 2: Watch Fireworks. By John Markoff
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Lance FortnowGeorgia Institute of Technology A Personal view of P versus NP
Step 1: Post Elusive Proof. Step 2: Watch Fireworks. • By John Markoff • … VinayDeolalikar, 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
$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
We can efficiently find a matching even among millions of men and women avoiding having to search all the possibilities. Efficient Algorithms P
Given a solution to a clique problem we can check it quickly Efficiently Verifiable NP
Easy to Solve Easy to Verify P NP P and NP
P = NP Every problem we can verify efficiently we can solve efficiently
P ≠ NP There are problems we can verify quickly that we can’t solve quickly
? P = NP Can we solve every problem quickly if the solutions are easily verifiable?
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? The P versus NP Problem
Formalizing the Turing Machine Start State State Space Accept State Input Alphabet Tape Alphabet Blank Symbol Transition Function
Transition function • (state, symbol) →(state, symbol, direction) • Nondeterministic • Can map to multiple possibilities Transitions
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? Defining P and NP
Instead of Turing machine • Multiple tapes • Random access • λ – calculus • C++ • LaTeX • Probabilistic and Quantum computers might not define the same class Mathematically RoBUST
reductions A B
Hardest problems in NP • Cook-Levin 1971 • Boolean Formula Satisfiability NP-complete
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
WE CURE CANCER What happens if P = NP?
William of Ockham, English Franciscan Friar Occam’s Razor (14th Century) Entia non sunt multiplicanda praeter necessitatem Occam’s Razor
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
Rosetta Stone • 196 BC Decree in three languages • Greek • Deomotic • Hieroglyphic • In 1822, Jean-François Champollion found a simple grammar. Translation
How do you deal with NP-completeness? Dealing with Hardness
Brute Force Heuristics Small Parameters Approximation Solve a Different Problem Give Up Dealing with Hardness
