Computer Assisted Proof ofComputer Assisted Proof ofOptimal Approximability Optimal Approximability

ResultsResults

Uri ZwickUri Zwick

Tel Aviv University

SODA’02, January 6-8,SODA’02, January 6-8,San Francisco San Francisco

Optimal approximability results require the proof of some nasty

real inequalities

Computerized proof of real inequalities

The The MAX 3-SATMAX 3-SAT problemproblem

487354

4353762

543652321

xxxxxx

xxxxxxx

xxxxxxxxx

Random assignment1/2

LP-based algorithm3/4Yannakakis ’94

GW ’94

SDP-based algorithm? 7/8 ?Karloff, Zwick ’97

The The MAX 3-CSPMAX 3-CSP problemproblem

),,(),,(),,(

),,(),,(),,(

),,(),,(),,(

754987387547

843675357624

543365223211

xxxfxxxfxxxf

xxxfxxxfxxxf

xxxfxxxfxxxf

Random assignment1/8

SDP-based algorithm? 1/2 ?Zwick ’98

Hardness resultsHardness results (FGLSS ’90, AS ’92, ALMSS ’92,

BGS ’95, Raz ’95, Håstad ’97)

Ratio for MAX 3-SAT P=NP7

8

1

2Ratio for MAX 3-CSP P=NP

Probabilistically Checkable Probabilistically Checkable ProofsProofs

PROOF

VERIFIER

CLAIM (xL)

RA

ND

OM

BIT

S

PCPc,s(log n , 3)

PCP1-ε,½(log n , 3) = NP(Håstad ’97)

PCP1-ε,½-ε(log n , 3) = P(Zwick ’98)

A Semidefinite A Semidefinite Programming Relaxation Programming Relaxation

of of MAX 3-SATMAX 3-SAT (Karloff, Zwick ’97)

0 0

0

4 ( ) ( ) 4 ( ) ( ),

4 44 ( ) ( )

, 14

, , || || 1 , 1

Max

s.t.

ijk ij

i j k j i kijk ijk

k i jijk i

k

jk

nn i i i i

v v v v v v v vz z

v v v vz z

v v v R v

w

i

z

n

Random hyperplane Random hyperplane roundingrounding

(Goemans, Williamson ’95)(Goemans, Williamson ’95)

v0

vi

vj

The probability that a clause xixjxk is satisfied

ij

iv

jv

0v

kv

is equal to the volume of a certain spherical

tetrahedron

SphericalSpherical volumes in volumes in SS33

1

4

2

3

θ12

λ13

2341312 ij

ij

Vol

),...,,(

Schläfli (1858) :

Spherical volume Spherical volume inequalities inequalities II

0

0

0

8

12031302

13022301

12032301

2

231303120201

coscoscoscos

coscoscoscos

coscoscoscos

whenever

),,,,,(Vol

Spherical volume Spherical volume inequalities inequalities IIII

0

0

0

832

7

13022301

13021203

12032301

2

12032301

2

231303120201

coscoscoscos

coscoscoscos

coscoscoscos

whenever

)coscoscoscos(

),,,,,(Vol

Computer Assisted Computer Assisted ProofsProofs

• The 4-color theorem

• The Kepler conjecture

A Toy ProblemA 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.

Idea of ProofIdea 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.

““Outline” of proofOutline” 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 < 0All that remains is to prove the claim on the boundary of the region.

How do we show that F’(x,y)≥0, for x0 ≤ x ≤ x1 , y0 ≤ y ≤y1

Interval Arithmetic

!!!

Interval ArithmeticInterval Arithmetic(Moore ’66)(Moore ’66)

A method of obtaining rigorous numerical results, in spite of the inherently inexact floating point arithmetic used.

yx yx

yx

IEEE-754 floating point standard

Interval ArithmeticInterval ArithmeticBasic Arithmetical Basic Arithmetical

OperationsOperations

},,,max{

},,,min{

],[*],[

],[],[],[

11011000

11011000

1010

11001010

yxyxyxyx

yxyxyxyx

yyxx

yxyxyyxx

Interval ArithmeticInterval ArithmeticInterval extension of elementary Interval extension of elementary

functionsfunctions

Let f(x) be a real function. If X is an interval, then let f(X) = { f(x) | xX }.

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.

The The “Fundamental “Fundamental Theorem”Theorem” of Interval of Interval

ArithmeticArithmetic

),(),( YXFYXf

)exp(

cossin),(

yx

xyyxyxf

)(),(

YXEXP

XCOSYYSINXYXF

Easy to implement using operator overloading

TheThe RealSearchRealSearch 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.

Concluding RemarksConcluding Remarks

• What is a proof?

• Need for general purpose tools( Numerica, GlobSol, RealSearch RealSearch ))

• Is there a simple proof?