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Computer Assisted Proof of Optimal Approximability Results

Computer Assisted Proof of Optimal Approximability Results. Uri Zwick Tel Aviv University SODA’02, January 6-8, San Francisco. Optimal approximability results require the proof of some nasty real inequalities. Computerized proof of real inequalities. The MAX 3-SAT problem.

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Computer Assisted Proof of Optimal Approximability Results

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  1. Computer Assisted Proof ofOptimal Approximability Results Uri ZwickTel Aviv University SODA’02, January 6-8,San Francisco

  2. Optimal approximability results require the proof of some nasty real inequalities Computerized proof of real inequalities

  3. The MAX 3-SAT problem

  4. The MAX 3-CSP problem

  5. Hardness results(FGLSS ’90, AS ’92, ALMSS ’92,BGS ’95, Raz ’95, Håstad ’97) Ratio for MAX 3-SAT P=NP Ratio for MAX 3-CSP P=NP

  6. PROOF VERIFIER RANDOM BITS CLAIM (xL) Probabilistically Checkable Proofs PCP1-ε,½(log n , 3) = NP(Håstad ’97) PCP1-ε,½-ε(log n , 3) = P(Zwick ’98) PCPc,s(log n , 3)

  7. A Semidefinite Programming Relaxation of MAX 3-SAT(Karloff, Zwick ’97)

  8. Random hyperplane rounding(Goemans, Williamson ’95) vi vj v0

  9. The probability that a clause xixjxk is satisfied is equal to the volume of a certain spherical tetrahedron

  10. Sphericalvolumes in S3 4 Schläfli (1858) : 3 λ13 2 1 θ12

  11. Spherical volume inequalities I

  12. Spherical volume inequalities II

  13. Computer Assisted Proofs • The 4-color theorem • The Kepler conjecture

  14. A Toy Problem • Show that F(x,y)≥0, for 0 ≤ x,y ≤ 1. • F(x,y) is “complicated”. • F(x,y) ≥ F’(x,y), where F’(x,y) is “simple”. • ∂F(x,y)/∂x and ∂F(x,y)/∂yare “simple”. • F(0,0)=0.

  15. Idea of Proof Show, somehow, that the claim holds on the boundary of the region. It is then enough to show that F’(x,y)≥ 0, at critical points, i.e., at points that satisfy∂F(x,y)/∂x = ∂F(x,y)/∂y= 0.

  16. “Outline” of proof • Partition [0,1]2 into rectangles, such that in each rectangle, at least one of the following holds: • F’(x,y) ≥ 0 • ∂F(x,y)/∂x > 0 • ∂F(x,y)/∂x < 0 • ∂F(x,y)/∂y > 0 • ∂F(x,y)/∂y < 0 All that remains is to prove the claim on the boundary of the region.

  17. How do we show thatF’(x,y)≥0, for x0 ≤ x ≤ x1 ,y0 ≤ y ≤y1 Interval Arithmetic!!!

  18. Interval Arithmetic(Moore ’66) A method of obtaining rigorous numerical results, in spite of the inherently inexact floating point arithmetic used. IEEE-754 floating point standard

  19. Interval ArithmeticBasic Arithmetical Operations

  20. Interval ArithmeticInterval extension of elementary functions Let f(x) be a real function. If Xis an interval, then let f(X) = { f(x) | xX }. An interval function F(X) is an interval extension of f(x) if f(X)  F(X), for every X. It is not difficult to implement interval extensions SIN, COS, EXP, etc., of sin, cos, exp, etc.

  21. The “Fundamental Theorem” of Interval Arithmetic Easy to implement using operator overloading

  22. TheRealSearchsystem A very naïve system that uses interval arithmetic to verify that given collections of real constraints have no feasible solutions. Used to verify the spherical inequalities needed to obtain proofs of the 7/8 and 1/2conjectures.

  23. Concluding Remarks • What is a proof? • Need for general purpose tools( Numerica, GlobSol, RealSearch ) • Is there a simple proof?

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