Christian Sohler | 03.11.10 Every Property of Hyperfinite Graphs is Testable Ilan Newman and...

Christian Sohler | 03.11.10

Every Property of Hyperfinite Graphs is TestableIlan Newman and Christian Sohler

Christian Sohler | Dortmund, 17.02.10

Prof. Dr. Christian SohlerKomplexitätstheorie und effiziente Algorithmen

Property Testing

Property Testing [Rubinfeld, Sudan, 96; Goldreich, Goldwasser, Ron, 98]

Oracle access to huge object O Decide whether O has a property P or is far-away from P Goal: Perform this relaxed decision task in sublinear or even constant time Algorithm may be randomized

Definition An object is e-far from P, if it differs in more than an e-fraction of its

description from any object with property P

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The Bounded-Degree Graph Model

Bounded degree graph model [Goldreich, Ron, 02] Graph G=(V,E) with maximum degree d (bounded-degree graph) Oracle access:

(a) random vertex

(b) i-th edge incident to vertex v

Definition (Graph Property; e-far) A graph property is a family of graphs P= P(i), where each P(i) is a set of

graphs on i vertices that is closed under isomorphism. An n-vertex graph G=(V,E) is e-far from a property P, if it differs from any

bounded-degree graph in P by more than edn edges.

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Testable Graph Properties

Definition (testable property) A graph property P is called non-uniformly testable in the bounded-degree

graph model, if there is a function q=q(e,d) such that for any n there is an algorithm A(e,d,n) that on input an n-vertex graph G with maximum degree at most d(a) makes at most q queries to G,(b) accepts G with probability at least 2/3, if G has property P, and(c) rejects G with probability at least 2/3, if G is e-far from P.

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Hyperfinite Graphs

Definition (hyperfiniteness) A graph G is (e,k)-hyperfinite, if one can remove en edges from G such that

only connected components of at most k vertices remain. A family F of graphs is r-hyperfinite, if for any e>0 every graph GF is (e,r(e))-hyperfinite.

Example Planar graphs are r-hyperfinite

for some suitable function r by the planar separator theorem

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Property Testing in Hyperfinite Graphs / of Hyperfinite PropertiesPrevious work Every hereditary property is testable in hyperfinite graphs

[Czumaj, Shapira,Sohler ,09] Every monotone hyperfinite property is testable

[Benjamini, Schramm, Shapira, 08] Every hereditary hyperfinite property is testable [Hassidim, Kelner, Nguyen,

Onak, 09] Every property is testable in graphs of bounded growth [Elek 07]

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Our results

This work [Newman, S., 11] Every property is testable in hyperfinite graphs. Every hyperfinite graph property is testable. Hyperfinite graph isomorphism is testable. Many graph parameters are estimable. The distribution of k-disks of two hyperfinite bounded-degree graphs is close,

iff the graphs are close.

Definition (k-disk) A k-disk of a vertex v is the subgraph rooted at v that is induced by all vertices

at distance at most k from v.

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Some further related work

Motifs A motif is a subgraph that occurs more frequently than in a random graph. Can be used to characterize graphs that come from certain domains

„The motifs shared by ecological food webs were distinct from the motifs shared by the genetic networks of Escherichia coli and Saccharomyces cerevisiae or from those found in the World Wide Web.” R. Milo, S. Shen-Orr, S. Itzkovitz, N. Kashtan, D. Chklovskii, U. Alon. Network Motifs: Simple Building Blocks of Complex Networks. Science, Vol. 298. no. 5594, pp. 824 - 827, 2002.

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Constant size graph representations

Definition For an input graph G, let G‘ be an (e,k)-representative of G, if

(a) G‘ has connected components of size at most k(b) G‘ can be obtained from G by removing at most edn edges

Observation If k is independent of n, then G‘ has a constant size representation: List all D(k) graphs on k vertices H(1), H(2), …, H(D(k)) Take a D(k)-dimensional vector whose i-th entry is the number of

occurences of isomorphic copies of H(i) in G‘

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A Property Tester

Idea: Reduce property testing in hyperfinite graphs to graphs whose components

have constant size

The Tester: Let P= P(i) be an arbitrary graph property Let P‘(n) = {H‘ : HP(n) and H‘ is an ( /2e ,k)-representative of H} Assume we know an ( /2e ,k)-representative G‘ of G Accept, iff G‘P‘(n)

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Local Partitioning oracles

Local Planar Partitioning Oracle [Hassidim, Kelner, Nguyen, Onak, 09] Provides „local access“ to a partition that has connected components of size at most k(e) and has been obtained by removing, say, at most edn/4 edges from G „Local access“ means that only some k-disks are sampled from G

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Local Partitioning oracles

GlobalPartitioning(k,d) [Hassidim, Kelner, Nguyen, Onak, 09] p = (p ,…, p ) = random permutation of the vertices P = while G is not empty do Let v be the first vertex in G according to p if there exists a (k,d)-isolated neighborhood of v in G then S = this neighborhood else S = {v} P = P {S} remove vertices in S from the graph

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1 n

(k,d)-isolated neighborhood of v:Connected set S of size at most k with vS and at most d|S| edges between S and V



How to obtain the representative of a graph?

Algorithm for computing the representative: Estimate the frequency of connected components by uniform sampling using

the planar partitioning oracle (take size of component into account)

Analysis: Chernoff bounds

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Second result

Result: The distribution of k-disks of two hyperfinite bounded-degree graphs is close,

iff the corresponding graphs are close.

Sketch of proof: „<=“: easy „=>“: We know that every property of hyperfinite graphs is testable So, for fixed n-vertex graph consider the property of being H Let ALG be our property testing algorithm for this property On input H, ALG must accept w.p. at least 2/3 We claim that, if a graph H‘ has similar distribution of k-disks as H, then the

tester does not reject w.p. > 1/2 => H‘ is e-close to H

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Proof of claim

Claim If H‘ has a similar distribution on k-disks as H, then the tester on input H‘ does

not reject w.p.>1/2

Sketch of Proof: The result of the local partitioning algorithm depends on the sampled k-disks

(and internal randomness) If the distribution of k-disks is similar, then the behaviour of the algorithms

should be similar unless the k-disks intersect However, as n goes to infinity, this probability tends to 0

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Summary

Testable graph properties (bounded-degree model) Every hyperfinite property Properties defined by the local structure (for example, triangle freeness) Certain combinations of the above (e.g. union) Other properties, like connectivity…

Open Questions Characterize ‚remaining‘ testable properties Property testing in expander graphs?

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Thank you!

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Christian Sohler | 03.11.10 Every Property of Hyperfinite Graphs is Testable Ilan Newman and...

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