L á szl ó Lov á sz Eötvös Lor ánd University Budapest May 2013 1 Happy Birthday Ravi May 2013 2 Cut norm of matrix A n x n The Weak Regularity Lemma ID: 614094
Download Presentation The PPT/PDF document "Algorithms on large graphs" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
Slide1
Algorithms on large graphs
László Lovász Eötvös Loránd University, Budapest
May 2013
1
Happy Birthday Ravi!Slide2
May 20132
Cut norm
of
matrix A
n
x
n
:
The Weak Regularity Lemma
Cut
distance of two
graphs with
V
(
G
) = V(G’):
(extends to edge-weighted)Slide3
May 20133
The Weak Regularity LemmaAvereged graph GP (P partition of V(
G))
1
1/2
Template graph
G/
P
1
1/2
1/2
1
0
0
2/5
2/5
1/5Slide4
May 2013
4The Weak Regularity LemmaFor every graph G and every >0 there isa partition with and
Frieze – Kannan 1999Slide5
May 20135
Algorithms for large graphs- Graph is HUGE.- Not known explicitly, not even
the
number of nodes.
Idealize
:
define
minimum
amount
of
info
.
How is the
graph given?Slide6
May 20136
Dense case: cn2 edges. - We can sample a uniform random node a bounded number of times, and see edges between sampled nodes. „Property testing”, constant time algorithms: Arora-Karger-Karpinski
, Goldreich
-Goldwasser-
Ron
,
Rubinfeld-Sudan
,
Alon-Fischer-Krivelevich-Szegedy,
Fischer
, Frieze-
Kannan, Alon-Shapira
Algorithms for large graphsSlide7
Computing a structure: find a maximum cut, regularity partition,...
Computing a structure: find a maximum cut, regularity partition,...May 20137Algorithms for large graphs Parameter estimation: edge density, triangle density, maximum cut
Property testing: is the graph bipartite? triangle-free?
perfect?
Computing a constant
size encoding
The partition (cut,...) can be
computed
in polynomial time.
For every node, we can determine
in constant time
which class
it belongs toSlide8
May 20138
Representative set Representative set of nodes: bounded size, (almost) every node is “similar” to one of the nodes in the setWhen are two nodes similar? Neighbors? Same neighborhood?Slide9
May 20139
This is a metric, computable in the sampling model
Similarity
distance
of
nodes
s
t
v
w
uSlide10
May 201310
Representative set Strong representative set U: for any two nodes in
s,
tU
,
d
sim
(
s
,
t
) > for
all nodes s, d
sim(U,s)
Average representative set U: for any two nodes s,
t
U
,
d
sim
(
s
,
t
) >
for
a random
node
s, Ed
sim(U,s) 2
Slide11
May 2013
11Representative sets and regularity partitionsIf P = {S1, . . . , Sk
} is a weak regularity partition
with error , then we
can select nodes
v
i
S
i
such that
S = {v1, . . . , vk} is an average
representative set with error <
4.If SV is an average representative set with error
, then the Voronoi cells of S form a weak regularity partition with error < 8.
L-SzegedySlide12
May 201312
Voronoi
diagram
=
weak
regularity
partition
Representative
sets and
regularity
partitionsSlide13
May 201313
Every graph has an average representative setwith at most nodes.
Representative sets
If
S
V
(
G
) and
d
sim
(
u,v
)> for all u,vS, then
Every
graph
has
a
strong
representative
set
with
at
most
nodes
.
AlonSlide14
May 201314
Example: every average representative sethas nodes.
Representative sets
angle
dimension
1/
Slide15
May 201315
Representative sets and regularity partitionsFrieze-Kannan
For every graph
G
and
>0 there are
u
i
,
v
i
{0,1}
V
(G) and ai
such thatSlide16
May 201316
Construct weak representative set UHow to compute a (weak) regularity partition?
Each
node is i
n
same
class
as
closest representative.Slide17
May 201317
- Construct representative set- Compute weights in template graph (use sampling)- Compute max cut in template graph
How
to
compute
a maximum
cut
?
(Different algorithm implicit by Frieze-
Kannan
.)
Each
node is on same side as
closest representative.Slide18
May 201318
Given a bigraph with bipartition {U,W} (|U|=|W|=n)and c[0,1], find a maximum subgraph with all degreesat most c|U|.How to compute
a maximum matching?Slide19
Nondeterministically estimable parameters
Divine help: coloring the nodes, orienting and coloring the edgesg: parameter defined on directed, colored graphsg’(H)=max{g(G): G’=H}; shadow of g
G
:
directed, (edge)-colored graph
G
’:
forget orientation, delete some colors,
forget coloring
;
shadow of G
f
nondeterministically estimable: f=
g’,where g is an estimable parameter of colored directed graphs.
May 201319Slide20
Examples: density of maximum cut
May 201320the graph contains a subgraph G’ with all degrees cn and |E(G’)| an2edit distance from a testable property
Fischer- Newman
Goldreich-Goldwasser-Ron
Nondeterministically
estim
able p
arametersSlide21
Every nondeterministically estimable graph
pproperty is testable.L-VesztergombiN=NP for dense
property
testing
Every nondeterministically
estim
able graph
p
aratemeter
is
estim
able.
L-Vesztergombi
Proof
via graph limit theory:p
ure
existence proof
of an algorithm...
May 2013
21
Nondeterministically
estim
able p
arametersSlide22
May 201322
More generally, how to compute a witness in non-deterministic property testing?How to compute a maximum matching?Slide23
May 201323
Happy Birthday Ravi!