Transcript A perfect notion László Lovász Microsoft Research

To the memory of Claude Berge
A perfect notion
László Lovász
Microsoft Research
[email protected]
Noisy channels
Alphabet {u,v,w,m,n}
u
can be confused
v
n
w
m
Largest safe subset: {u,m}
 (G )  max # of independent nodes
But if we allow words...
Safe subset: {uu,nm,mv,wn,vw}
 (G 2 )
Shannon capacity of G:
(G)  lim k  (G k )
k 
G2 :
Shannon 1956
Trivial: (G )   (G)
For which graphs does (G)=(G) hold?
Which are the minimal graphs for which
(G)>(G)?
Sufficient for equality: G can be covered by (G) cliques.
 (G )   (G )
Min-max theorems for graphs
 (G ) : stability number
 (G )  n   (G ) : node-cover number
 (G )   (G ) : clique number
 (G ) : chromatic number
 (G ) : matching number
(G )  n   (G ) : edge-cover number
 '(G ) : chromatic index
 (G ) : maximum degree
 (G )   (G )
 (G )   (G )  (G )
 (G )   (G)
 (G )   '(G )
Three theorems of König:
For bipartite graphs G:
For their linegraphs H:
 '(G )   (G )
 (H )  (H )
 (G )   (G)
 (G )  (G )
 ( H )  ( H )
[  (G)]
More...
Interval graphs satisfy  (G )   (G ) Gallai
Interval graphs satisfy  (G )   (G)
Hajós
Every cycle is triangulated   (G )   (G ) Hajnal-Surányi
Every cycle is triangulated   (G )   (G) Berge
Comparability graphs satisfy  (G )   (G )
Dilworth
Comparability graphs satisfy  (G )   (G)
Every odd cycle is triangulated   (G )   (G ) Gallai
What is common?
- condition is inherited by induced subgraphs
- theorems come in pairs
Perfect graph: every induced subgraph H
satisfies (H)=(H)
Weak perfect graph conjecture:
The complement of a perfect graph is perfect.
Fulkerson 1970
LL 1971
Strong perfect graph conjecture:
G is perfect  neither G nor its complement
contains an odd cycle
Chudnovsky
Robertson
Seymour
Thomas 2002
G is perfect

 (G ') (G ')  n
for all induced subgraphs G’
LL 1972
 Perfectness is in co-NP
Is it in NP? or P?
YES! Chudnovsky
Cornuejols
Liu
Seymour
Vušković
Hypergraphs
G
H : V ( H )  {cliquesof G}
E ( H )  {nodes of G}
 (G )   (G)
 ( H )   '( H )
 (G )   (G )
(H )   (H )
for all induced
subgraphs
for all partial
subhypergraphs
What are “bipartite” hypergraphs?
Berge, Fournier, Las Vergnas, Erdős, Hajnal, L
Antiblocking polyhedra
Fulkerson 1971
(polarity in the nonnegative orthant)
K
n

convex corner
( K * )*  K ;
K *  {x 
n

: xT y  1 y  K }
facets of K  vertices of K *
The stable set polytope
 A : incidence vector of set A
STAB(G)  conv{ A : A stable set in G }
Defined through vertices –
how to describe by
facets/linear inequalities?
Finding valid inequalities for STAB(G)
xi  0
i V (G )
xi  x j  1 ij  E (G )
 xi  1
sufficient iff
G is bipartite
 clique B
i B
| C | 1
xi 
 odd hole C

2
i C
sufficient iff
G is t-perfect
Chvátal
sufficient iff
G is perfect
More formulations:
G is perfect

STAB(G )*  STAB(G )
FSTAB(G )  {x 
V (G )

:  xi  1 clique B in G }
i B
G is perfect

STAB(G )  FSTAB(G)
Geometric representation of graphs
and semidefinite optimization
Orthogonal representation:
i V
a
vi  ¡
d
 | ui | 1
 ui u j  0 (ij  E )
T
Trivial…
b
a=b=c
c
a
d
e
0
c=d
Less trivial…
C5
in
3
Profile of a geometric representation:
x  ( xi : i V (G )), xi  (ui )12
TH(G)={profiles of ONR’s of G }
FSTAB(G)
TH(G)
STAB(G)
Grötschel
Lovász
Schrijver
xi : i V (G)
xi 2  xi
i V (G )
xi x j  0

of a stable set
ij  E (G )
linearize...
x0  1
x is the incidence vector
Yij  xi x j
Y00  1
Yii  Yi 0
Yij  0
i V (G )
ij  E(G)
rk(Y )  1 Y positive semidefinite
(Y )10isthe
TH(incidence
G)
 (Y)
vector of a stable set
TH(G ) * TH(G )
G is perfect

TH(G )  STAB(G )  FSTAB(G )
 “Weak” conjecture

TH(G ) is a polytope
One can maximize
a linear function over TH(G)
in polynomial time
 semidefinite
optimization
For a perfect graph, (G), (G) can be
computed in polynomial time.
Graph entropy
Körner 1973
p: probability distribution on V(G)
 p log x : x STAB(G)}

iV (G )
H (G, p)  min{
i
i
  p log x
i
H ( Kn , p)  H ( p)   pi log pi
i
 const
t
1
H (G, p)  limt  minU V (Gt ) log  (G [U ])
t
Pt (U ) .99
connected iff distinguishable
Want: encode most of V(G)t by 0-1 words of min length,
so that distinguishable words get different codes.
(measure of “complexity” of G)
H ( F , p)  H (G, p)  H ( F  G, p)
H (G, p )  H (G, p )  H ( p )
p : H (G, p )  H (G , p )  H ( p )

G is perfect
Csiszár, Körner, Lovász, Marton, Simonyi
Nullstellensatz - Positivestellensatz
the following system is unsolvable (in )
xi 2  xi
 (G )  k 
xi x j  0
x
i
i V (G )
ij  E (G )
 k 1
  polynomials i ,  ij , 
2

(
x
 i i  xi )    ij xi x j  (k  1   xi )  1
Useless...
the conditions
xi 2  xi
xi x j  0
 (G )  k 
i V (G )
ij  E (G )
imply
x
i
k
  polynomials i ,  ij , t
2
2
x

k


(
x

x
)


x
x


 i
 i i i  ij i j  t
G is perfect

 weights wi  0 (i V (G ))
 linear polynomials t and scalars ai , bij , such that
2
2
a
x


(
G
)

a
(
x

x
)

b
x
x


 ii w
 i i i  ij i j  t
xi 2  xi
i V (G )
xi x j  0
ij  E (G )
i
j

x is the incidence
vector
of a stable set
(1  xi )(1  x j )  0
1  xi  x j  0
(1  x1 )(1  x2  x3 )  0
1
2
5
x1 (1  x3  x4 )  0
(1  x1 )(1  x4  x5 )  0
4
3
2  x1  x2  x3  x4  x5  0
1
3
2
(1 other
x1 )(1derivations:
 x2  x3  x4 )  0
Two
4
1  x1  x2  x3  x4  0
(1  x1 )(1  x2 )(1  x3 )(1  x4 )  0
(1  x1  x2  x3  x4 )2  0
1 linear
x1  xinequality
2  x3  x4  0
In at most n steps, every
valid for STAB(G) can be derived this way.
LL-Schrijver
(trivial)
edge constraints
edge constraints
odd hole constraints
LL-Schrijver
edge+
odd hole constraints
?
clique constraints
?
edge+
triangle constraints
?
Every such constraint is supported on a
subgraph with at most one degree >4.
Lipták
What we discussed...
And what else we should have...
0-error capacity Shannon
Min-max theorems for
bipartite graphs König
rigid circuit graphs, comparability graphs
Gallai, Dilworth, Berge,...
Perfect graphs - 2 conjectures Berge
Hypergraphs - bipartite and König
Balanced, 2-colorable,...
Berge
The stable set polytope and antiblocking
Blocking polyhedra
Fulkerson, Chvátal
Graph entropy
Körner; Csiszár, Körner, Lovász, Marton, Simonyi
Geometric representation and semidefinite optimization Approximation algorithms
Grötschel, Lovász, Schrijver
Lift-and-cut
Nullstellensatz - Positivestellensatz
Game theory Berge, Duchet, Boros, Gurevich
Structure theory Chvátal, Chudnovsky, Cornuejols, Liu, Robertson,
Seymour, Thomas, Vušković