Algorithms on large graphs
László Lovász
Eötvös Loránd University, Budapest
May 2013
1
The Weak Regularity Lemma
Cut norm of matrix A
A
W
=
1
n
2
nxn:
å å
[ ]
m ax
S ,T Í n
iÎ S
A ij
jÎ T
Cut distance of two graphs with V(G) = V(G’):
d W (G , G ') =
1
n
2
m a x | e G ( S ,T ) - e G ' ( S ,T ) |
S ,T Í V ( G )
(extends to edge-weighted)
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2
The Weak Regularity Lemma
Avereged graph GP (P partition of V(G))
1/2
1
0
Template graph G/P
1/2
1
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2/5
2/5
1/2
1
0
1/5
3
The Weak Regularity Lemma
For every graph G and every >0 there is
a partition
| P |= 2
with
O (1/ e 2 )
and d W (G , G P ) < e
Frieze – Kannan 1999
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4
Algorithms for large graphs
How is the graph given?
- Graph is HUGE.
- Not known explicitly, not even the number of
nodes.
Idealize: define minimum amount of info.
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5
Algorithms for large graphs
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
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6
Algorithms for large graphs
Parameter estimation: edge density, triangle density,
maximum cut
Property testing: is the graph bipartite? triangle-free?
perfect?
Computing a structure:
find a maximum
Computing
a constantcut,
regularity
partition,...
size
encoding
The partition (cut,...) can be
computed in polynomial time.
For every node, we can determine
in constant time which class
it belongs to
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7
Representative set
Representative set of nodes: bounded size,
(almost) every node is “similar” to
one of the nodes in the set
When are two nodes similar? Neighbors?
Same neighborhood?
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8
Similarity distance of nodes
d sim ( s , t ) := E v E u ( a su avu ) - E w ( a tw aw v )
s
w
u
t
v
This is a metric, computable in the sampling model
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Representative set
Strong representative set U:
for any two nodes in s,tU, dsim(s,t) > 
for all nodes s, dsim(U,s)  
Average representative set U:
for any two nodes s,tU, dsim(s,t) > 
for a random node s, Edsim(U,s)  2
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10
Representative sets and regularity partitions
If P = {S1, . . . , Sk} is a weak regularity partition
with error , then we can select nodes viSi
such that S = {v1, . . . , vk} is an average
representative set with error < 4.
If SV is an average representative set with error ,
then the Voronoi cells of S form a weak regularity
partition with error < 8.
L-Szegedy
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11
Representative sets and regularity partitions
Voronoi diagram
= weak regularity partition
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12
Representative sets
Every graph has an average representative set
with at most 2
O (1/ e 2 )
nodes.
Every graph has a strong representative set
with at most 2
O (lo g (1/ e ) / e 2 )
nodes.
Alon
If S  V(G) and dsim(u,v)> for all u,vS, then
S = 2
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O (lo g (1/ e ) / e 2 )
13
Representative sets
Example: every average representative set
has 2
W(1/ e 2 )
nodes.
dimension 1/
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angle 
14
Representative sets and regularity partitions
For every graph G and >0 there are
ui, vi {0,1}V(G) and ai  such that
k
AG -
å
i= 1
aiu iv
T
i
< e,
W
æ1 ö
k = O çç 2 ÷
÷
÷
çè e ø
Frieze-Kannan
d sim ( s , t ) := E v E u ( a su avu ) - E w ( a tw aw v )
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15
How to compute a (weak) regularity partition?
Construct weak representative set U
Each node is in same class as closest representative.
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16
How to compute a maximum cut?
- Construct representative set
- Compute weights in template graph (use sampling)
- Compute max cut in template graph
Each node is on same side as closest representative.
(Different algorithm implicit by Frieze-Kannan.)
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17
How to compute a maximum matching?
Given a bigraph with bipartition {U,W} (|U|=|W|=n)
and c[0,1], find a maximum subgraph with all degrees
at most c|U|.
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18
Nondeterministically estimable parameters
Divine help: coloring the nodes,
orienting and coloring the edges
G: directed, (edge)-colored graph
G’: forget orientation, delete some colors,
forget coloring; shadow of G
g: parameter defined on directed, colored graphs
g’(H)=max{g(G): G’=H}; shadow of g
f nondeterministically estimable: f=g’,where g is an
estimable parameter of colored directed graphs.
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19
Nondeterministically estimable parameters
Examples: density of maximum cut
Goldreich-Goldwasser-Ron
edit distance from a testable property
Fischer- Newman
the graph contains a subgraph G’ with
all degrees cn and |E(G’)| an2
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20
Nondeterministically estimable parameters
Every nondeterministically estimable graph
paratemeter is estimable.
L-Vesztergombi
Every nondeterministically
estimable graph
N=NP for dense
property testing
pproperty is testable.
L-Vesztergombi
Proof via graph limit theory:
pure existence proof
of an algorithm...
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21
How to compute a maximum matching?
More generally, how to compute a witness in
non-deterministic property testing?
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23