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A perfect notion

To the memory of Claude Berge. A perfect notion. L á szl ó Lov á sz Microsoft Research lovasz@microsoft.com. u. can be confused. v. n. m. w. Noisy channels. Alphabet { u,v,w,m,n }. Largest safe subset: { u,m }. But if we allow words. Safe subset: { uu,nm,mv,wn,vw }.

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A perfect notion

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

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

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

  4. Trivial: Sufficient for equality: G can be covered by (G) cliques. Shannon 1956 For which graphs does (G)=(G) hold? Which are the minimal graphs for which (G)>(G)?

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

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

  7. Gallai Interval graphs satisfy Interval graphs satisfy Hajós Hajnal-Surányi Every cycle is triangulated  Berge Every cycle is triangulated  Dilworth Comparability graphs satisfy Comparability graphs satisfy Gallai Every odd cycle is triangulated  More...

  8. Perfect graph: every induced subgraph H satisfies (H)=(H) What is common? - condition is inherited by induced subgraphs - theorems come in pairs Weak perfect graph conjecture: The complement of a perfect graph is perfect. Fulkerson 1970 LL 1971 Chudnovsky Robertson Seymour Thomas 2002 Strong perfect graph conjecture: G is perfect  neither G nor its complement contains an odd cycle

  9. G is perfect  for all induced subgraphs G’ LL 1972  Perfectness is in co-NP Is it in NP? or P? YES!Chudnovsky Cornuejols Liu Seymour Vušković

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

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

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

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

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

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

  16. b a=b=c c a d e 0 c=d Trivial…

  17. Less trivial…

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

  19. xis the incidence vector of a stable set linearize... • (Y)1is the incidence vector of a stable set Y positive semidefinite

  20.  “Weak” conjecture 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.

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

  22. 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)

  23. Csiszár, Körner, Lovász, Marton, Simonyi

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

  25. the conditions imply

  26. G is perfect

  27. i j 1 2 5 4 3 xis the incidence vector of a stable set

  28. 1 2 Two other derivations: 3 4 In at most n steps, every linear inequality valid for STAB(G) can be derived this way. LL-Schrijver

  29. edge constraints odd hole constraints LL-Schrijver ? edge+ odd hole constraints ? edge+ triangle constraints clique constraints ? Every such constraint is supported on a subgraph with at most one degree >4. Lipták (trivial) edge constraints

  30. And what else we should have... Balanced, 2-colorable,... Blocking polyhedra Approximation algorithms Lift-and-cut Game theoryBerge, Duchet, Boros, Gurevich Structure theory Chvátal,Chudnovsky, Cornuejols, Liu, Robertson, Seymour, Thomas, Vušković What we discussed... 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

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