1. Eötvös Loránd Tudományegyetem, Budapest
The many facets of the
Regularity Lemma
Lovász László
Eötvös Loránd Tudományegyetem, Budapest
Cim
May 2012

2. The Lemma Original version
Given  >0
The nodes of any graph can be partitioned
into a small number
of essentially equal parts
so that
most bipartite graphs between 2 parts
are essentially random
(with different densities).
difference at most 1
with  k2 exceptions
for subsets X,Y of parts Vi,Vj
# of edges between X and Y
is pij|X||Y| ±  (n/k)2
May 2012

3. The Lemma Weaker and Stronger
Original Regularity Lemma Szemerédi 1976
“Weak” Regularity Lemma Frieze-Kannan 1999
“Strong” Regularity Lemma Alon-Fisher-Krivelevich
-M.Szegedy 2000
Tao 2006
L-B.Szegedy 2007
May 2012

4. The Lemma Weak version S
pij: density between Vi and Vj
May 2012

5. The Lemma Weak version replace edges between Vi and Vj
by random edges with density pij
to get G'
May 2012

6. The Lemma Original versionVi X Y Vj
May 2012

7. The Lemma Strong version
May 2012

8. Regularity lemma in pictures Pixel pictures AG G WG
May 2012

9. Regularity lemma in pictures Pixel pictures
May 2012

10. Regularity lemma in pictures Pixel pictures
essentially random
Nodes can be so ordered
May 2012

11. Removal Lemma A great application
>0 ’>0
# of triangles ’n3 
we can delete n2 edges to get a triangle-free graph.
Ruzsa - Szemerédi
Implies: the r3(n) theorem,...
Dependence of 1/' on 1/ is at most
tower of height log(1/)
Fox
May 2012

12. Counting Lemma Subgraph densities
t(F,G): Probability that random map V(F)V(G)
preserves edges
(Can be defined for weighted graphs G)
|t(F,G) - t(F,H)|  |E(F)| □(G,H)
If □(G,H) is small, then G and G’ are similar
in many other respects...
May 2012

13. Linear algebra version Low rank approximation
Cut norm of nxn matrix A: :
Frieze - Kannan
May 2012

14. Analytic version Approximation in Hilbert space
May 2012

15. Analytic version 2. Distance of graphs
cut distance
(a) V(G) = V(G')
(b) |V(G)| = |V(G')|
(c) |V(G)| =n, |V(G')|=m
(blow up nodes, or fractional overlay)
May 2012

16. Analytic version 2. Distance of graphs
“Weak" Regularity Lemma (approximation form):
May 2012

17. Analytic version Graphons
W0 = {W: [0,1]2 [0,1], symmetric, measurable}
"graphon"
t(F,WG) = t(F,G): Probability that random map
V(F)V(G) preserves edges
May 2012

18. Analytic version 2. Distance of functions
May 2012

19. Analytic version 2. Approximation by stepfunctions
“Weak" Regularity Lemma for graphons:
May 2012

20. Analytic version 2 Approximation by stepfunctions
“Strong" Regularity Lemma:
May 2012

21. Analytic version 3 Compactness
Strongest (but non-effective) Regularity Lemma:
1-step
Martingale
Convergence
Theorem
k-step
May 2012

22. Analytic version 2. Deriving the strong version
1-step
3-step
2-step
-neighborhoods
with radius εk
L1-neigborhoods
with radius ε0
May 2012

23. Geometric version Similarity metricw u
This is a metric
Representative set U:
for any two nodes in U, dsim(s,t) > 
for most nodes, dsim(U,v)  
May 2012

24. = weak regularity partition
Geometric version Representative set and regularity
Voronoi diagram
= weak regularity partition
May 2012

25. Geometric version Representative set and regularity
If P = {S1, , Sk} is a partition of V(G) such that
d(G,GP) = , then we can select nodes viSi
such that the average similarity distance from
S = {v1, , vk} is at most 4.
If SV and the average similarity distance from S is ,
then the Voronoi cells of S form a partition P such that
d(G,GP) 8.
L-Szegedy
May 2012

26. Geometric version Regularity and dimension
Every graph can be partitioned into
sets with similarity diameter <.
Alon
Graph with similarity distance
is almost finite dimensional
May 2012

27. Algorithm What answer to expect?
- Cannot list for all nodes!
Input: We can sample a uniform random node
a bounded number of times,
and see edges between sampled nodes.
Output: For any given node, we want to tell
in which class it belongs to
(after some preprocessing)
May 2009

28. Algorithm Computing a weak regularity partition
Construct representative set U
sim(s,t) is computable in the sampling model
Each node is in same class as closest representative.
May 2012

29. Congratulations Endre!
And finally...
Congratulations Endre!
May 2012