1. Lance Fortnow Georgia Institute of Technology

2. 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, 2010 From: Deolalikar, Vinay To: 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…

3. $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

15. We can efficiently find a matching even among millions of men and women avoiding having to search all the possibilities. P

21. Given a solution to a clique problem we can check it quickly NP

22. Easy to Solve Easy to Verify NPP

23. P = NP

24. P NP

25. P = NP ?

30. Transition Function Tape Alphabet Blank Symbol Input Alphabet State Space Start State Accept State

31. Transition function (state, symbol) (state, symbol, direction) Nondeterministic Can map to multiple possibilities

33. Instead of Turing machine Multiple tapes Random access λ – calculus C++ LaTeX Probabilistic and Quantum computers might not define the same class

34. A B

35. Hardest problems in NP Cook-Levin 1971 Boolean Formula Satisfiability

37. 1935: Turings 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 100s of NP- complete problems Now thousands of NP-complete problems over many disciplines

39. WE CURE CANCER

41. William of Ockham, English Franciscan Friar Occams Razor (14 th Century) Entia non sunt multiplicanda praeter necessitatem

42. William of Ockham English Franciscan Friar Occams Razor (14 th Century) Entities must not be multiplied beyond necessity The simplest explanation is usually the best. If P = NP we can find that simplest explanation.

43. Rosetta Stone 196 BC Decree in three languages Greek Deomotic Hieroglyphic In 1822, Jean-François Champollion found a simple grammar.

48. How do you deal with NP-completeness?

49. Brute Force Heuristics Small Parameters Approximation Solve a Different Problem Give Up

52. 123456 S1S1 InOutInOutIn S2S2 OutInOut InOut S3S3 Out S4S4 InOutInOutInOut S5S5 In S6S6 OutInOut In

53. NP doesnt 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 n 1.8 time.

54. Measure complexity by size of circuit. Different circuits for each input length. Efficient computation essentially equivalent to small circuits.

55. 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.

56. ( 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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