A perfect notion László Lovász Microsoft Research To the memory of Claude Berge
Noisy channels Alphabet {u,v,w,m,n} u n m w v can be confused Largest safe subset: {u,m}
But if we allow words... Safe subset: {uu,nm,mv,wn,vw} Shannon capacity of G:
For which graphs does ( G )= ( G ) hold? Shannon 1956 Trivial: Which are the minimal graphs for which (G)> (G)? Sufficient for equality: G can be covered by ( G ) cliques.
Min-max theorems for graphs matching number clique number stability number edge-cover number chromatic number node-cover number chromatic index maximum degree
Three theorems of König: For bipartite graphs G : For their linegraphs H :
Interval graphs satisfy Hajós Every cycle is triangulated Hajnal-Surányi Comparability graphs satisfy Dilworth Every odd cycle is triangulated Gallai Interval graphs satisfy Gallai Every cycle is triangulated Berge Comparability graphs satisfy More...
What is common? - condition is inherited by induced subgraphs Weak perfect graph conjecture: The complement of a perfect graph is perfect. Strong perfect graph conjecture: G is perfect neither G nor its complement contains an odd cycle Fulkerson 1970 LL 1971 Chudnovsky Robertson Seymour Thomas theorems come in pairs Perfect graph: every induced subgraph H satisfies ( H )= ( H )
Perfectness is in co-NP Is it in NP? or P?YES! Chudnovsky Cornuejols Liu Seymour Vušković G is perfect for all induced subgraphs G’ LL 1972
Hypergraphs for all induced subgraphs for all partial subhypergraphs What are “bipartite” hypergraphs? Berge, Fournier, Las Vergnas, Erdős, Hajnal, L
Antiblocking polyhedra Fulkerson 1971 convex corner (polarity in the nonnegative orthant)
The stable set polytope Defined through vertices – how to describe by facets/linear inequalities?
sufficient iff G is bipartite sufficient iff G is perfect Finding valid inequalities for STAB(G) sufficient iff G is t-perfect Chvátal
More formulations: G is perfect G is perfect
Geometric representation of graphs and semidefinite optimization Orthogonal representation:
a d e b c c=d a=b=c 0 Trivial…
Less trivial…
FSTAB(G) TH(G) Profile of a geometric representation: STAB(G) Grötschel Lovász Schrijver TH(G)= {profiles of ONR’s of }
x is the incidence vector of a stable set linearize... (Y) 1 is the incidence vector of a stable set Y positive semidefinite
One can maximize a linear function over TH(G) in polynomial time For a perfect graph, ( G ), ( G ) can be computed in polynomial time. “Weak” conjecture semidefinite optimization
Graph entropy Körner 1973 p : probability distribution on V(G)
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 )
Csiszár, Körner, Lovász, Marton, Simonyi
Nullstellensatz - Positivestellensatz Useless... the following system is unsolvable (in )
the conditions imply
G is perfect
x is the incidence vector of a stable set ij
Two other derivations: In at most n steps, every linear inequality valid for STAB(G) can be derived this way. LL-Schrijver
(trivial)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
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 Berge The stable set polytope and antiblocking Fulkerson, Chvátal Graph entropy Körner; Csiszár, Körner, Lovász, Marton, Simonyi Geometric representation and semidefinite optimization Grötschel, Lovász, Schrijver Nullstellensatz - Positivestellensatz What we discussed... Balanced, 2-colorable,... Structure theory Chvátal, Chudnovsky, Cornuejols, Liu, Robertson, Seymour, Thomas, Vušković Blocking polyhedra Approximation algorithms Lift-and-cut And what else we should have... Game theory Berge, Duchet, Boros, Gurevich