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Every Property of Hyperfinite Graphs is Testable Ilan Newman and Christian Sohler Christian Sohler | 03.11.10 Prof. Dr. Christian Sohler Komplexitätstheorie 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 Christian Sohler | Dortmund, 17.02.10 2 Prof. Dr. Christian Sohler Komplexitätstheorie und effiziente Algorithmen 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. Christian Sohler | Dortmund, 17.02.10 3 Prof. Dr. Christian Sohler Komplexitätstheorie und effiziente Algorithmen 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. Christian Sohler | Dortmund, 17.02.10 4 Prof. Dr. Christian Sohler Komplexitätstheorie und effiziente Algorithmen 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 Christian Sohler | Dortmund, 17.02.10 5 Prof. Dr. Christian Sohler Komplexitätstheorie und effiziente Algorithmen Property Testing in Hyperfinite Graphs / of Hyperfinite Properties Previous 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] Christian Sohler | Dortmund, 17.02.10 6 Prof. Dr. Christian Sohler Komplexitätstheorie und effiziente Algorithmen 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. Christian Sohler | Dortmund, 17.02.10 7 Prof. Dr. Christian Sohler Komplexitätstheorie und effiziente Algorithmen 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. Christian Sohler | Dortmund, 17.02.10 8 Prof. Dr. Christian Sohler Komplexitätstheorie und effiziente Algorithmen 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‘ Christian Sohler | Dortmund, 17.02.10 9 Prof. Dr. Christian Sohler Komplexitätstheorie und effiziente Algorithmen 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 (e/2,k)-representative of H} Assume we know an (e/2,k)-representative G‘ of G Accept, iff G‘P‘(n) Christian Sohler | Dortmund, 17.02.10 10 Prof. Dr. Christian Sohler Komplexitätstheorie und effiziente Algorithmen 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 Christian Sohler | Dortmund, 17.02.10 11 Prof. Dr. Christian Sohler Komplexitätstheorie und effiziente Algorithmen Local Partitioning oracles GlobalPartitioning(k,d) [Hassidim, Kelner, Nguyen, Onak, 09] p = (p1 ,…, pn ) = 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 (k,d)-isolated neighborhood of v: else S = {v} Connected set S of size at most k P = P {S} with vS and at most d|S| edges remove vertices in S from the graph between S and V Christian Sohler | Dortmund, 17.02.10 12 Prof. Dr. Christian Sohler Komplexitätstheorie und effiziente Algorithmen 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 Christian Sohler | Dortmund, 17.02.10 13 Prof. Dr. Christian Sohler Komplexitätstheorie und effiziente Algorithmen 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 Christian Sohler | Dortmund, 17.02.10 14 Prof. Dr. Christian Sohler Komplexitätstheorie und effiziente Algorithmen 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 Christian Sohler | Dortmund, 17.02.10 15 Prof. Dr. Christian Sohler Komplexitätstheorie und effiziente Algorithmen 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? Christian Sohler | Dortmund, 17.02.10 16 Prof. Dr. Christian Sohler Komplexitätstheorie und effiziente Algorithmen Thank you! Christian Sohler | Dortmund, 17.02.10 17