1. Computer Assisted Proof of Optimal Approximability Results Uri Zwick Uri Zwick Tel 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 1/2Random assignment Yannakakis ’94 GW ’94 3/4LP-based algorithm Karloff, Zwick ’97? 7/8 ?SDP-based algorithm

4. The MAX 3-CSP problem 1/8Random assignment Zwick ’98? 1/2 ?SDP-based algorithm

5. Hardness results 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. Probabilistically Checkable Proofs PROOF VERIFIER CLAIM (x  L) RANDOM BITS PCP c,s (log n, 3) PCP 1-ε,½ (log n, 3) = NP (Håstad ’97) PCP 1-ε,½- ε (log n, 3) = P (Zwick ’98)

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

8. Random hyperplane rounding (Goemans, Williamson ’95) v0v0 vivi vjvj

9. The probability that a clause x i  x j  x k is satisfied is equal to the volume of a certain spherical tetrahedron

10. Spherical volumes in S 3 1 4 2 3 θ 12 λ 13 Schläfli (1858) :

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)/∂y are “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 that F’(x,y)≥0, for x 0 ≤ x ≤ x 1, y 0 ≤ y ≤y 1  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 Arithmetic Basic Arithmetical Operations

20. Interval Arithmetic Interval extension of elementary functions Let f(x) be a real function. If X is 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. The RealSearch The RealSearch system 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 1/2 conjectures.

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