Algorithms on large graphs

László Lovász

Eötvös Loránd University, Budapest

May 2013 1

Happy Birthday Ravi!

2May 2013

[ ]2 ,

1max ijS T n

i S j T

A An Í

Î Î

= å åW

Cut norm of matrix Anxn:

The Weak Regularity Lemma

'2 , ( )

1( , ') max | ( , ) ( , ) |G G

S T V Gd G G e S T e S T

n Í= -W

Cut distance of two graphs with V(G) = V(G’):

(extends to edge-weighted)

3May 2013

The Weak Regularity Lemma

Avereged graph GP (P partition of V(G)) 11/2

Template graph G/P11/2

1/210

0

2/5

2/5

1/5

4May 2013

The Weak Regularity Lemma

For every graph G and every >0 there is

a partition with

and 2(1/ )| | 2O e=P ( , )d G G e<PW

Frieze – Kannan 1999

5May 2013

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?

May 2013 6

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 graphs

Computing a structure: find a maximum cut, regularity partition,...Computing a structure: find a maximum cut, regularity partition,...

May 2013 7

Algorithms 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 becomputed in polynomial time.

For every node, we can determine in constant time which class

it belongs to

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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?

May 2013 9

sim( , ) : E E ( ) E ( )v u su vu w wvtwa a ad t as = -

This is a metric, computable in the sampling model

Similarity distance of nodes

st

v

wu

May 2013 10

Representative set

Strong representative set U:

for any two nodes in s,tU, dsim(s,t) >

for all nodes s, dsim(U,s)

Average representative set U:

for any two nodes s,tU, dsim(s,t) >

for a random node s, Edsim(U,s) 2

May 2013 11

Representative sets and regularity partitions

If P = {S1, . . . , Sk} is a weak regularity partition with error , then we can select nodes viSi

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-Szegedy

May 2013 12

Voronoi diagram= weak regularity

partition

Representative sets and regularity partitions

May 2013 13

Every graph has an average representative set

with at most nodes. 2(1/ )2O e

Representative sets

If S V(G) and dsim(u,v)> for all u,vS, then

2(log(1/ / ))2OS e e=

Every graph has a strong representative set

with at most nodes. Alon

2(log(1/ / ))2O e e

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Example: every average representative set

has nodes. 2(1/ )2 eW

Representative sets

angle dimension 1/

May 2013 15

Representative sets and regularity partitions

Frieze-Kannan

12

1,

kT

G i i ii

A a u v k Oe

e=

æ ö÷ç ÷ç ÷ç- <è ø

=åW

For every graph G and >0 there are ui, vi {0,1}V(G) and ai such that

sim( , ) : E E ( ) E ( )v u su vu w wvtwa a ad t as = -

May 2013 16

Construct weak representative set U

How to compute a (weak) regularity partition?

Each node is in same class as closest representative.

May 2013 17

- 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.

May 2013 18

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|.

How to compute a maximum matching?

Nondeterministically estimable parameters

Divine help: coloring the nodes, orienting and coloring the edges

g: parameter defined on directed, colored graphs

g’(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 2013 19

Examples: density of maximum cut

May 2013 20

the graph contains a subgraph G’ with all degrees cn and |E(G’)| an2

edit distance from a testable property

Fischer- Newman

Goldreich-Goldwasser-Ron

Nondeterministically estimable parameters

Every nondeterministically estimable graph

pproperty is testable.L-Vesztergombi

N=NP for denseproperty testing

Every nondeterministically estimable graph

paratemeter is estimable.L-Vesztergombi

Proof via graph limit theory:pure existence proof

of an algorithm...

May 2013 21

Nondeterministically estimable parameters

May 2013 22

More generally, how to compute a witness in

non-deterministic property testing?

How to compute a maximum matching?

May 2013 23

Happy Birthday Ravi!