1. A perfect notion László Lovász Microsoft Research lovasz@microsoft.com To the memory of Claude Berge

2. Noisy channels Alphabet {u,v,w,m,n} u n m w v can be confused Largest safe subset: {u,m}

3. But if we allow words... Safe subset: {uu,nm,mv,wn,vw} Shannon capacity of G:

4. 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.

5. Min-max theorems for graphs matching number clique number stability number edge-cover number chromatic number node-cover number chromatic index maximum degree

6. Three theorems of König: For bipartite graphs G : For their linegraphs H :

7. 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...

8. 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 2002 - theorems come in pairs Perfect graph: every induced subgraph H satisfies  ( H )=  ( H )

9.  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

10. Hypergraphs for all induced subgraphs for all partial subhypergraphs What are “bipartite” hypergraphs? Berge, Fournier, Las Vergnas, Erdős, Hajnal, L

12. Antiblocking polyhedra Fulkerson 1971 convex corner (polarity in the nonnegative orthant)

13. The stable set polytope Defined through vertices – how to describe by facets/linear inequalities?

14. sufficient iff G is bipartite sufficient iff G is perfect Finding valid inequalities for STAB(G) sufficient iff G is t-perfect Chvátal

15. More formulations: G is perfect  G is perfect 

16. Geometric representation of graphs and semidefinite optimization Orthogonal representation:

17. a d e b c c=d a=b=c 0 Trivial…

18. Less trivial…

19. FSTAB(G) TH(G) Profile of a geometric representation: STAB(G) Grötschel Lovász Schrijver TH(G)= {profiles of ONR’s of }

20.  x is the incidence vector of a stable set linearize...  (Y) 1 is the incidence vector of a stable set Y positive semidefinite

21. 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

22. Graph entropy Körner 1973 p : probability distribution on V(G)

23. 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 )

24. Csiszár, Körner, Lovász, Marton, Simonyi

25. Nullstellensatz - Positivestellensatz Useless... the following system is unsolvable (in  )

26. the conditions imply

27. G is perfect 

28.  x is the incidence vector of a stable set ij 1 2 5 43

29. 3 21 4 Two other derivations: In at most n steps, every linear inequality valid for STAB(G) can be derived this way. LL-Schrijver

30. (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

31. 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