1. Hyperfinite graphs and combinatorial optimization László Lovász
Hungarian Academy of Sciences
and
Eötvös Loránd University, Budapest
January 2018

2. Don in Epidauros 1995
January 2018

3. Bounded degree ( D) Borel graph
Graphing, definition
Bounded degree ( D) Borel graph
on standard probability space V (say [0,1])
with “double counting” condition:
Extends to measure  on Borel subsets of V2.
„edge measure”
January 2018

4. „The notion of an algorithm is basic to
Why graphings?
„The notion of an algorithm is basic to
all of computer programming, so we should
begin with a careful analysis of this concept.”
Donald Knuth
January 2018

5. Find a perfect matching in G.
Why graphings?
Find a perfect matching in G.
distributed algorithm: Nguyen and Onak
January 2018

6. distributed algorithms
Why graphings?
distributed algorithms
local algorithms (on bounded degree graphs)
January 2018

7. distributed algorithms
Why graphings?
distributed algorithms
local algorithms (on bounded degree graphs)
property testing (on bounded degree graphs)
January 2018

8. distributed algorithms
Why graphings?
distributed algorithms
local algorithms (on bounded degree graphs)
property testing (on bounded degree graphs)
local algorithms on graphings
The infinite is a good approximation
of the large finite.
January 2018

9. Why graphings? Ergodic theory Finitely generated groups
Limits of convergent graphs sequences
with bounded degree
January 2018

10. Graphing, examples
January 2018

11. Graphing, examples unit circumference  irrational components:
2-way infinite paths
January 2018

12. , irrational, lin. indep 1x1 torus , irrational
Graphing, examples
unit circumference
, irrational, lin. indep  
1x1 torus
, irrational
components: grids  
components: grids
January 2018

13. Penrose tilings
rhombic icosahedron
de Bruijn - Bárász
January 2018

14. Local equivalence, definition
xV  Gx: component of x
uniform random x  Gx: random connected
rooted (countable)
graph
unimodular random
network
Benjamini - Schramm
January 2018

15. Local equivalence, definition
xV  Gx: component of x
G1, G2 locally equivalent:
for random uniform xV,
distributions of (G1)x and (G2)x
are the same
January 2018

16. 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.
January 2018

17. Local isomorphism, example 
(x,y)  x+y mod 1   
components: grids
components: grids
January 2018

18. 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
January 2018

19. Graph partition problem
k-edge-separator: TE(G), component of G-T
has  k nodes
Graphing G hyperfinite: sepk(G)0 (k)
January 2018

20. Hyperfinite graphings, examples 
Diophantine approximation  
January 2018

21. Hyperfinite graph families
Family G of finite graphs is hyperfinite:
Hyperfinite: paths, trees, planar graphs,
every non-trivial minor-closed property
Non-hyperfinite: expanders
January 2018

22. Every graph property is testable in any family of hyperfinite graphs.
Hyperfinite graph families
Every graph property is testable in any family
of hyperfinite graphs.
Many graph properties are
polynomial time testable for
graphs with bounded tree-width.
Newman – Sohler
(Benjamini-Schramm-Shapira,
Elek)
January 2018

23. {Gn} is hyperfinite  G is hyperfinite
Hyperfinite graph sequences
If Gn  G locally, then
{Gn} is hyperfinite  G is hyperfinite
Schramm
(Benjamini-Shapira-Schramm)
January 2018

24. ? Hyperfinite graphings If G1 and G2 are locally equivalent, then
G1 is hyperfinite  G2 is hyperfinite G1 G2 G ?
January 2018

25. Local isomorphism forward
Diophantine approximation 
(x,y)  x+y mod 1    
January 2018

26. Pushing forward and pulling back
measure
preserving
subset
linear
relaxation
measure
January 2018

27. Fractional graph partition problem
T: optimal k-edge-separator
January 2018

28. Fractional graph partition problem
probability distribution
„marginal” uniform
expected expansion
January 2018

29. Fractional graph/graphing partition problem
Define
Can be defined for
graphings
 probability distribution on Rk
with uniform marginal
no dependence on k
January 2018

30. Hyperfinite graphings
If G1 and G2 are locally equivalent, then
G1 is hyperfinite  G2 is hyperfinite G1 G2 G
January 2018

31. Algorithm: For j=1,2,..., select Y1,Y2,...k so that
Proof sketch
Algorithm: For j=1,2,..., select Y1,Y2,...k so that
Yj is the minimizer of
Output: X=Y1 Y2 ...
On a graphing:
no uncountable
sequence of steps!
Phases...
January 2018

32. Fractional separation duality
If G1 and G2 are locally equivalent, then G1 G2
easy ?
Duality!
January 2018

33. Fractional separation duality
Hahn-Banach + Riesz Representation
January 2018

34. Fractional separation duality
January 2018

35. Thanks, that’s all!
January 2018