1. Hyperfinite graphs and combinatorial optimization László Lovász
Hungarian Academy of Sciences
and
Eötvös Loránd University, Budapest
August 2017

2. Graphing, definition Bounded degree ( D) Borel graph on V=[0,1]
with “measure-preserving” condition:
Extends to measure  on Borel subsets of [0,1]2
„edge measure”
August 2017

3. Graphing, examples unit circumference  irrational components:
2-way infinite paths
August 2017

4. Graphing, examples unit circumference 1x1 torus , irrational
components: grids  
components: grids
August 2017

5. Graphing, examples ... ... ... ... 1 ... 1 ... connected to1
... 1
...
connected to
August 2017

6. Local equivalence, definition
xV  Gx: component of x
uniform random x  Gx: random connected
rooted (countable)
graph
= unimodular random
network
G1, G2 locally equivalent:
distributions of (G1)x and (G2)x
are the same
August 2017

7. Local isomorphism, definition
 : V(G1)  V(G2) local isomorphism:
measure preserving and
(x) isomorphism between
(G1)x and (G2)(x)
Existence of local isomorphism
proves local equivalence.
August 2017

8. Local isomorphism, example 
(x,y)  x+y mod 1   
components: grids
components: grids
August 2017

9. G1 and G2 are locally equivalent 
Local equivalence
G1 and G2 are locally equivalent 
G and local isomorphisms GG1, GG2. G1 G2 G
August 2017

10. Graph partition problem
k-edge-separator: TE(G), component of G-T
has  k nodes
August 2017

11. Hyperfinite (amenable) graphings
Graphing G hyperfinite: sepk(G)0 (k)
 k nodes
(T) small
August 2017

12. Hyperfinite graphings, examples1
...
not hyperfinite    
August 2017

13. ? Hyperfinite graphings If G1 and G2 are locally equivalent, then
G1 is hyperfinite  G2 is hyperfinite G1 G2 G ?
August 2017

14. Local isomorphism forward
(x,y)  x+y mod 1    
August 2017

15. Fractional graph partition problem
T: optimal k-edge-separator
August 2017

16. Fractional graph partition problem
probability distribution
„marginal” uniform
expected expansion
August 2017

17. Fractional graph partition problem
Define
Can be defined for
graphings
 probability distribution on Rk
with uniform marginal
no dependence on k
August 2017

18. Hyperfinite graphings
If G1 and G2 are locally equivalent, then
G1 is hyperfinite  G2 is hyperfinite G1 G2 G
August 2017

19. Hyperfinite graphings  
August 2017

20. Algorithm: For j=1,2,..., select Y1,Y2,... so that
Proof sketch
Algorithm: For j=1,2,..., select Y1,Y2,... so that
Yj is the minimizer of
Output: X=Y1 Y2 ...
On a graphing:
no uncountable
sequence of steps!
Phases...
August 2017

21. Fractional separation
If G1 and G2 are locally equivalent, then G1 G2
easy ?
Duality!
August 2017

22. Fractional separation, duality in finite case
Linear program:
Dual program:
August 2017

23. Fractional separation, duality in infinite case
August 2017

24. Fractional separation, duality in infinite case
Duality - Hahn-Banach
+ Riesz Representation
Compactification of graphings
August 2017

25. (I,A): standard Borel space : probability measure
Compact graphings
Graphing: (I,A,,E)
(I,A): standard Borel space
: probability measure
E: symmetric Borel subset of IxI
with measure-preserving condition
August 2017

26. Compact graphing: (J,A,,E) J: compact metric space A: Borel sets of J
Compact graphings
Compact graphing: (J,A,,E)
J: compact metric space
A: Borel sets of J
: probability measure
E: symmetric Borel subset of IxI
with measure-preserving condition, and
August 2017

27. Compact graphings    
not compact
compact
August 2017

28. Every graphing can be obtained from a
Compact graphings
Every graphing can be obtained from a
compact graphing by deleting components
of total measure zero.
August 2017

29. Fractional separation, duality in infinite case
August 2017

30. co-NP characterization of hyperfiniteness
Graphing G is not hyperfinite 
August 2017

31. Fractional separation
If G1 and G2 are locally equivalent, then
sep*(G1) = sep*(G2) G1 G2
easy ?
August 2017

32. Pushing forward and pulling back
measure
preserving
subset
linear
relaxation
measure
function
August 2017

33. Hyperfinite (amenable) graph families
Family G of graphs is hyperfinite:
Hyperfinite: paths, trees, planar graphs,
every non-trivial minor-closed property
Non-hyperfinite: expanders
August 2017

34. Hyperfinite graph sequences
Every locally convergent hyperfinite graph sequence
is locally-globally convergent.
Elek
Hatami – L – Szegedy
Every property of hyperfinite graphs is testable.
Newman – Sohler
(Benjamini-Schramm-Shapira,
Elek)
August 2017

35. Hyperfinite (amenable) graph families
O(1) nodes
o(n) edges
August 2017

36. {Gn} is hyperfinite  G is hyperfinite
If Gn  G locally, then
{Gn} is hyperfinite  G is hyperfinite
Schramm
Benjamini-Shapira-Schramm
August 2017

37. Thanks, that’s all!
August 2017

38. Combopt generalization
(V,H): hypergraph on n vertices, x {x}H
w: HR+, w({x})  1
August 2017

39. Combopt generalization
August 2017