computer assisted proof of optimal approximability results uri zwick uri zwick tel aviv university...
Post on 19-Dec-2015
217 views
TRANSCRIPT
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
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.
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.