Nondeterministic property testing

László LovászKatalin Vesztergombi

(k,G): labeled subgraph of G induced by

k random nodes.

Definitions

11 min( , ) ( , ) : ( ) ( ),d G H V H V G H Pd G P

September 2012 2

21( , )| ( ) ( ) |

| |

E G

VG G

Gd

E

P testable: there is a test property P’, such that

(a)for every graph G ∈ P and every k ≥ 1,

(k,G) ∈ P′ with probability at least 2/3, and

(b) for every ε > 0 there is a k0 ≥ 1 such that

for every graph G with d1(G,P) > ε

and every k ≥ k0 we have G(k,G) ∈ P′

with probability at most 1/3.

P: graph property

Testable graph properties

September 2012 3

Example: No edge.

Testable graph properties: examples

Example: All degrees ≤10.

Example: Contains a clique with ≥ n/2 nodes.

Example: Bipartite.

Example: Perfect.

September 2012 4

Removal Lemma:

’ if t(,G)<’, then we can delete n2 edges

to get a triangle-free graph.

Ruzsa - Szemerédi

G’: sampled induced subgraph

G’ not triangle-free G not triangle free

G’ triangle-free with high probability, G has few triangles

Example: triangle-free

Testable graph properties: examples

September 2012 5

Example: disjoint union of two isomorphic graphs

Testable graph properties: examples

Not testable!

September 2012 6

Every hereditary graph property is testable.

Alon-Shapira

inherited by induced subgraphs

Testable graph properties

September 2012 7

Nondeterministically testable graph properties

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

Q: property of directed, colored graphs

shadow(Q)={shadow(G): GQ};

G: directed, edge and node-colored graph

shadow(G): forget orientation, delete edges with certain colors, forget coloring

P nondeterministically testable: P= shadow(Q), where Q is a testable property of colored directed graphs.

September 2012 8

Examples: maximum cut contains ≥n2/100 edges

contains a clique with ≥ n/2 nodes

contains a spanning subgraph with a testable property P

we can delete ≤n2/100 edges to get a perfect graph

Nondeterministically testable graph properties

September 2012 9

Every nondeterministically testable graph

property is testable.

Main Theorem

„P=NP” for property testing in dense graphs

Pure existence proof of an algorithm

September 2012 10

L-V

Restrictions and extensions

Node-coloring can be encoded into the edge-coloring.

We will not consider orientation of edges.

Equivalent:

Certificate is given by unary and binary relations.

Ternary etc?

Theorem is false if functions are allowed besides

relations. (Example: union of two isomorphic graphs.)

September 2012 11

G

0 0 1 0 0 1 1 0 0 0 1 0 0 10 0 1 0 1 0 1 0 0 0 0 0 1 01 1 0 1 0 1 1 1 1 0 1 0 1 10 0 1 0 1 0 1 0 1 0 1 1 0 00 1 0 1 0 1 1 0 0 0 1 0 0 11 0 1 0 1 0 1 1 0 1 1 1 0 11 1 1 1 1 1 0 1 0 1 1 1 1 00 0 1 0 0 1 1 0 1 0 1 0 1 10 0 1 1 0 0 0 1 1 1 0 1 0 00 0 0 0 0 1 1 0 1 0 1 0 1 01 0 1 1 1 1 1 1 0 1 0 1 1 10 0 0 1 0 1 1 0 1 0 1 0 1 00 1 1 0 0 0 1 1 0 1 1 1 0 11 0 1 0 1 1 0 1 0 0 1 0 1 0

AG

WG

September 2012 12

From graphs to functions

September 2012 13

W0 = {W: [0,1]2 [0,1], symmetric, measurable}

Kernels and graphons

graph G graphon WG

W = {W: [0,1]2 , symmetric, bounded, measurable}

kernel

graphon

September 2012 14

'( , ' , ')) inf (

W Wd W WW W

'( , ') ( , )G GWG WG

( , ') 'Wd W WW W W

, [0,1]sup

S T S T

W W

W

There is a finite definition.

Cut distance

cut norm on L([0,1]2)

cut distance

A graph property P is testable iff for every sequence

(Gn) of graphs with |V(Gn)| and (Gn,P)0,

we have d1(Gn,P)0.

Cut distance and property testing

September 2012 15

L-Szegedy

September 2012 16

distribution of k-samplesis convergent for all k

Probability that random map V(F)V(G) preserves edges

(G1,G2,…) convergent: F t(F,Gn) is convergent

Convergence of a graph sequence

(G1,G2,…) convergent Cauchy in the cut distance

Borgs-Chayes-L-Sós-V

| ( )|

hom( , )

| ( ) |( , )

V F

F G

V Gt F G

September 2012 17

GnW : F t(F,Gn) t(F,W)

Limit graphon of a graph sequence

( , ) 0nGW WWd ®Equivalently:

( ) ( )[0,1]

( ,( , ) )Î

= ÕòV F

i jij E F

W x x dxt F W

September 2012 18

For every convergent graph sequence (Gn)

there is a WW0 such that GnW .

Conversely, W (Gn) such that GnW .

W is essentially unique

(up to measure-preserving transformation).

Limit graphon: existence and uniqueness

L-Szegedy

Borgs-Chayes-L

Let Gn be a sequence of graphs, and let U be a

graphon such that Gn U. Then the graphs Gn can be

labeled so that ‖ ‖ 0

nGW U

Convergence in norm

(Wn ): sequence of uniformly bounded kernels

with Wn 0. Then WnZ 0 for every integrable

function Z: [0,1]2 .

September 2012 19

Borgs-Chayes-L-Sós-V

L-Szegedy

1

( , )k

h h

h

U Wd

U W ‖ ‖

k-graphons

k-graphon: W=(W1,...,Wk), where W1,...,Wk0 and

W1+...+Wk=1

fractional k-coloration

111

( , )k

h h

h

d U W

U W ‖ ‖

September 2012 20

Sample (r,W): random x1,...,xr[0,1], connect i to j

with color c with probability Wc(xi,xj)

Ln: sequence of k-edge-colored graphs.

Ln convergent: distribution of (r,Ln) is convergent.

Convergence of k-graphons

September 2012 21

Ln convergent sequence of k-colored graphs

k-graphon W : (r,Ln)(r,W) in distribution.

Equivalently: ( , ) 0

nLW WWd ®

L-Szegedy

H1, H2, ...

in Q

shadow(Hn)=Gn

... J2, J1

shadow(Jn)=Fn

close to Q

G1, G2, ... ... F2, F1

in P far from P

Main Theorem: Proof

September 2012 22

Let W=(W1,...,Wk) be a k-graphon, and let .

Let Fn U. Then there exist k-colored graphs Jn on

V(Jn) = V(Fn) such that shadow(Jn) = Fn and Jn W.

1

mh

h

U W

Main Lemma

September 2012 23

September 2012 24

+

F

Proof (k=3, m=2)

W 1 W 2

24

+=

H 1 H 2

September 2012

(H1, H2) fractional edge-2-coloring (J1, J2) edge-2-coloring by randomization

Proof (cont)

1 1 2 2( , ), ( , )d H J d H JW W are small (Chernoff bound)

1 2

1 2( , ), ( , )H H

d W W d W WW W are small

Two things to prove:

25

1

1 1

0

1 1

0

11

0

1

( )

( )( )nn

HS T S

G

S T

F

T

FS T

U U

GS T

U

WW W W W

U

W W WW

U UW U

UW

W

U

September 2012 26

Proof (cont)

0

0 0

1

0

) ( ) 0(F F F F U

S T S T S TU U U

WW W W U W U

U

1 ‖ ‖

1 11 1

00

0( ) | |F F

S T UU

W WW WW W

U U U U

1 1

( ) ( )F F

S T

W WW U W U

U U

September 2012 27

Proof (cont)

September 2012 28

Sampling method: We can sample a uniform random

node a bounded number of times, and explore its

neighborhood to a bounded depth.

Bounded degree graphs (≤D)

September 2012 29

Maximum cut cannot be estimated in this model

(random D-regular graph vs. random bipartite D-regular graph)

PNP in this model

(random D-regular graph vs. union of two random D-regular graphs)

Bounded degree graphs (≤D)