Artur CzumajDept of Computer Science & DIMAP

University of Warwick

Testing Expansion in Testing Expansion in Bounded Degree GraphsBounded Degree Graphs

Joint work with Christian Sohler

Topic of this talkTopic of this talk

• How to distinguish between good expanders and weak expanders

• I will show that for graphs of bounded degree, we can distinguish between expanders and graphs that are “far” even from poor expanders in O(n1/2) time

[in the framework of property testing]

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Property testingProperty testing

• Classical decision problem:– Given a property P and input instance I– Does I has property P?

• What we want to study [relaxation]:– Is I close to satisfy property P?

Often it’s hard (NP-complete or even undecidable)

Can work fast even for NP-hard or undecidable properties

Property Testing definitionProperty Testing definition

• Given input x

• If x has the property tester passes

• If x is -far from any string that has the property tester fails

• error probability < 1/3

Notion ofNotion of -far-far depends on the problem;depends on the problem;Typically: one needs to changeTypically: one needs to change fraction of the input fraction of the input

to obtain object satisfying the propertyto obtain object satisfying the property

Typically we think aboutTypically we think about as on a small constant, say,as on a small constant, say, = 0.1 = 0.1

Graph propertiesGraph properties

• Measure of being far/close from a property• Is graph connected or is farfar from being connected?

These two graphs are These two graphs are closeclose to be connected to be connected

Graph propertiesGraph properties

• Measure of being far/close from a property• Is graph connected or is farfar from being connected?

far from being connected

11stst definition definition

Graph G is -far from satisfying property PIf one needs to modify more than -fraction of entries in adjacency matrixadjacency matrix to obtain a graph satisfying P

¢n2 edges have to be added/deleted

Suitable for dense graphsSuitable for dense graphs

Usually “boring” for sparse graphsUsually “boring” for sparse graphs

0 1 0 0 1

1 0 1 1 1

0 1 0 0 1

0 1 0 0 0

1 1 1 0 0

22ndnd definition definition

Graph G is -far from satisfying property PIf one needs to modify more than -fraction of entries in adjacency listsadjacency lists to obtain a graph satisfying P

Suitable for sparse graphsSuitable for sparse graphs

Main model: graphs of bounded degreeMain model: graphs of bounded degree

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5 2

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1 4 5 3

5

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Adjacency matrix modelAdjacency matrix model

• There are very fast property testers• They’re very simple

– Typical algorithm:• Select a random set of vertices U• Test the property on the subgraph induced by U

• The analysis is (often) very hard• We understand this model very well

– mostly because of very close relation to combinatorics

General resultGeneral result

• Every hereditary property can be tested in constant-timeconstant-time!

• Property is hereditaryhereditary if– It holds if we remove vertices

[Alon & Shapira, 2003-2005]

Adjacency matrix modelAdjacency matrix model

• There are very fast property testers• They’re very simple

– Typical algorithm:• Select a random set of vertices U• Test the property on the subgraph induced by U

• The analysis is (often) very hard• We understand this model very well

– mostly because of very close relation to combinatorics

What’s about adjacency lists model ?What’s about adjacency lists model ?

• We consider bounded-degree model– graph has maximum degree dd [constant]

• Much less is known

Constant time testing?Constant time testing?

• Very few properties known (for general graphs)– connectivity– k-connectivity– H-freeness– …– very few more

Bounded-degree adjacency list modelBounded-degree adjacency list model

• Recent result (C & Sohler, 2007):

– Any hereditary property is testable in Any hereditary property is testable in constant-time if the input graph belongs to constant-time if the input graph belongs to a hereditary and non-expanding family of a hereditary and non-expanding family of graphsgraphs

– Corollary: Testing hereditary properties in Corollary: Testing hereditary properties in planar graphs can be done in constant planar graphs can be done in constant time.time.But this doesn’t deal with general graphs

Bounded-degree adjacency list modelBounded-degree adjacency list model

• Testing bipartitness (2-colorability)– Can be done in O(nO(n1/21/2 / / O(1)O(1))) time (Goldreich &

Ron)– Cannot be done faster (Goldreich & Ron)

• So: no constant-time algorithms

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But we had O(1/O(1))-time tester in the adjacency matrix model

For general bounded degree graphs, testing most of natural properties requiresuperconstant-time (typically, (n1/2))

This talk: testing expansionThis talk: testing expansion

• Can we quickly test if a (bounded degree) graph has good expansion?

(n1/2) lower bound [Goldreich & Ron]– even to distinguish between a very good

expander and disconnected graph with several huge components

• Most property testing results in the bounded degree model use expansion

This talk: testing expansionThis talk: testing expansion

• Can we test if a (bounded degree) graph has good expansion in O(n1/2) time?

Algorithm of Goldreich and RonAlgorithm of Goldreich and Ron

• Choose s = O(1/) vertices at random• For each chosen vertex v

– run m = O(n1/2) random walks of length O(log n)– count the number of collisions at the end-

vertices– If the number of collisions is too large then

• STOP & Reject

• If no STOP then– acceptRandom walks are on regular graphs: for each node v:

choose a random neighbor with prob. 1 / 2 deg(v)otherwise stay

Algorithm of Goldreich and RonAlgorithm of Goldreich and Ron

• Key use of the well-known fact:

– If a graph is expander (regular) then random walk of length O(log n) will reach a random vertex

– If we run c n1/2 random walks (for an appropriate constant c) then we expect the number of collisions to be close to expected

• this is testing of uniform distribution

•Idea/hope:–If graph is not expander then for many starting

vertices random walk won’t “mix”

Key task – prove the following:If graph is -far from expander then for many starting vertices random walk won’t mix

In general: obviously wrong

Can graphs far from expanders rapidly Can graphs far from expanders rapidly mix?mix?

• We don’t understand well non-expanders• We understand even less graphs that are far

from expanders

• Goldreich and Ron suggested algorithm• Couldn’t analyze it• Gave a conjecture – which if true – would

yield property tester– Conjecture: quite deep property of graphs that

are far from being expander

Testing vertex expansionTesting vertex expansion

• Graph G = (V,E) is an -expander-expander ifFor every X 4V, |X||V|/2 holds: |N(X)| |X|

• Our goal:– Distinguish graphs with vertex expansion Distinguish graphs with vertex expansion fromfrom

those those -far from having vertex expansion -far from having vertex expansion **, , ** ¿¿

In our case * = O(/log n)

Goldreich & Ron analyzed algebraic notion of expansion

Algorithm of Goldreich and RonAlgorithm of Goldreich and Ron

• Choose s = O(1/) vertices at random• For each chosen vertex v

– run m = O(n1/2) random walks of length O(log n)– count the number of collisions at the end-

vertices– If the number of collisions is too large then

• STOP & Reject

• If no STOP then– accept

m 12 s n1/2/2 l 16 d2 ln(n/)/2

s 16/

(1+7) ( ) /nm2

Testing vertex expansionTesting vertex expansion

Key Property:

•If G is -far from *-expander then there is a set of vertices X 4 V such that

– |V|/4 |X| (1+)|V|/2– |N(X)| c* * |X|

Think:– G is -far from *-expander X has c c |X| / log n |X| / log n

neighbors

Small ratio cut Small ratio cut bad mixing bad mixing

• Think = (1)• What if we have set X with |N(X)| c|X|/log n ?

• Run a random walk of length < c log n/2 that starts at a random vertex from X

• With a constant probability it won’t leave X !

Small ratio cut Small ratio cut bad mixing bad mixing

Start random walk at a random node at VSuppose it starts at a node at X Until it’s in X, in each step it has “probability” |N(X)|/|X| of “leaving” X

If random walk is shorter than |X|/|N(X)| we don’t expect to leave X

Collision probability will be largeWe’ll reject!We’ll reject!

X has small neigborhood

X V - X

Small ratio cut Small ratio cut bad mixing bad mixing

• We have a large (of size |V|/4) set X with small neighborhood

• With a constant probability a node from X will be a starting node for random walks

• With a constant probability, we will have too many collisions for such a node

• With a constant probability we will REJECT

It suffices to prove “Key Property”It suffices to prove “Key Property”

Key Property:

•If G is -far from *-expander then there is a set of vertices X 4 V such that

– |V|/4 |X| (1+)|V|/2– |N(X)| c* * |X|

Auxiliary lemmaAuxiliary lemma

If G=(V,E) has A 4 V with |A| n /4 such that G[V – A] is an c*- expander then G is not -far from *-expander

If G is If G is -far from -far from **-expander:-expander:every “small” set can be removed so that every “small” set can be removed so that

the remaining graph is still not an expanderthe remaining graph is still not an expander

Auxiliary lemmaAuxiliary lemma

If G=(V,E) has A 4 V with |A| n /4 such that G[V – A] is an c*- expander then G is not -far from *-expander

We can modify We can modify dn/2 edges in G dn/2 edges in G

to obtain an to obtain an **-expander-expander

A V – Ac *-expander

Auxiliary lemmaAuxiliary lemma

If G=(V,E) has A 4 V with |A| n /4 such that G[V – A] is an c*- expander then G is not -far from *-expander

We can modify We can modify dn/2 edges in G dn/2 edges in G

to obtain an to obtain an **-expander-expander

1.1. Remove all edges incident to ARemove all edges incident to A2.2. Add (d-1)-regular good expander in AAdd (d-1)-regular good expander in A3.3. Remove a matching M of size |A|/2 in G[V-A]Remove a matching M of size |A|/2 in G[V-A]4.4. Add arbitrary matching between A and MAdd arbitrary matching between A and M

Proving “Key Property”Proving “Key Property”

If G=(V,E) has A 4 V with |A| n /4 such that G[V – A] is an c*- expander then G is not -far from *-expander

If G is If G is -far from -far from **-expander:-expander:every “small” set can be removed so that every “small” set can be removed so that

the remaining graph is still not an expanderthe remaining graph is still not an expander

1.1. Start with X = Start with X = ;;2.2. G[V-A] is not an expander G[V-A] is not an expander

99 A A 4 V-X with small neighborhood V-X with small neighborhood3.3. X = A X = A [[ X X

4.4. Repeat step 2 with new A until |X| Repeat step 2 with new A until |X| |V| /4 |V| /4

Proves “Key Property”

SummarizingSummarizing

• We can distinguish between graphs (of maximum degree d) that have -vertex expansion and are -far from graph with (c/log n)-vertex expansion in time

O(d2 ln(n/) n1/2/(2 3))

Open questions:Open questions:

Can we distinguish (in O(n1/2) time) between graphs that have -vertex expansion and are -far from graph with /c-vertex expansion?

Same question for algebraic definition of expansion

Partial answer (Kale & Seshadhri’2007):Partial answer (Kale & Seshadhri’2007):O(nO(n1/21/2)-time to distinguish between )-time to distinguish between graphs of max-degree d that have graphs of max-degree d that have -vertex expansion and -vertex expansion and those with max-degree those with max-degree 2d2d and and -far from graphs -far from graphs

with with /c-vertex expansions/c-vertex expansions