Testing Expansion in Bounded-Degree Graphs

We consider the problem of testing expansion in bounded degree graphs. We focus on the notion of vertex-expansion: an alpha-expander is a graph G = (V, E) in which even-subset U sube V of at most |V|/2 vertices has a neighborhood of size at least alphaldr|U|. Our main result is that one can distinguish good expanders from graphs that are far from being weak expanders in time O tilde(radicn). We prove that the property testing algorithm proposed by Goldreich and Ron (2000) with appropriately set parameters accepts every alpha-expander with probability at least 2/3 and rejects every graph that is epsiv-far from an alpha*-expander with probability at least 2/3, where alpha*=Theta(alpha2/(d2log (n/epsiv))) and d is the maximum degree of the graphs. The algorithm assumes the bounded-degree graphs model with adjacency list graph representation and its running time is O(d2(radicn log (n/epsiv))/alpha2epsiv3).

[1]  Dana Ron,et al.  Property Testing in Bounded Degree Graphs , 2002, STOC '97.

[2]  Roger B. Myerson,et al.  Optimal Auction Design , 1981, Math. Oper. Res..

[3]  H. Moulin.  Incremental cost sharing: Characterization by coalition strategy-proofness , 1999 .

[4]  László Lovász,et al.  Graph limits and parameter testing , 2006, STOC '06.

[5]  Avrim Blum,et al.  Near-optimal online auctions , 2005, SODA '05.

[6]  Kenji Obata,et al.  A lower bound for testing 3-colorability in bounded-degree graphs , 2002, The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings..

[7]  E. Fischer.  THE ART OF UNINFORMED DECISIONS: A PRIMER TO PROPERTY TESTING , 2004 .

[8]  Artur Czumaj,et al.  Sublinear-Time Algorithms , 2006, Bull. EATCS.

[9]  Richard P. McLean,et al.  Optimal Selling Strategies under Uncertainty for a Discriminating Monopolist When Demands Are Interdependent , 1985 .

[10]  Mark Jerrum,et al.  The Markov chain Monte Carlo method: an approach to approximate counting and integration , 1996 .

[11]  Zoë Abrams,et al.  Revenue maximization when bidders have budgets , 2006, SODA '06.

[12]  Alistair Sinclair,et al.  Improved Bounds for Mixing Rates of Markov Chains and Multicommodity Flow , 1992, Combinatorics, Probability and Computing.

[13]  Noga Alon,et al.  Eigenvalues and expanders , 1986, Comb..

[14]  Mark Jerrum,et al.  Approximate Counting, Uniform Generation and Rapidly Mixing Markov Chains , 1987, International Workshop on Graph-Theoretic Concepts in Computer Science.

[15]  Anna R. Karlin,et al.  Truthful and Competitive Double Auctions , 2002, ESA.

[16]  D. Ron,et al.  Testing the diameter of graphs , 2002 .

[17]  Felix Wu,et al.  Incentive-compatible online auctions for digital goods , 2002, SODA '02.

[18]  Oded Goldreich,et al.  Combinatorial property testing (a survey) , 1997, Randomization Methods in Algorithm Design.

[19]  Jeremy I. Bulow,et al.  The Simple Economics of Optimal Auctions , 1989, Journal of Political Economy.

[20]  Anna R. Karlin,et al.  Optimization in the private value model: competitive analysis applied to auction design , 2003 .

[21]  Vijay Kumar,et al.  Online learning in online auctions , 2003, SODA '03.

[22]  Noga Alon,et al.  A Characterization of the (Natural) Graph Properties Testable with One-Sided Error , 2008, SIAM J. Comput..

[23]  Ronitt Rubinfeld,et al.  Algorithms column: sublinear time algorithms , 2003, SIGA.

[24]  Dana Ron,et al.  A Sublinear Bipartiteness Tester for Bounded Degree Graphs , 1999, Comb..

[25]  Artur Czumaj,et al.  On testable properties in bounded degree graphs , 2007, SODA '07.

[26]  N. Linial,et al.  Expander Graphs and their Applications , 2006 .

[27]  Andrew V. Goldberg,et al.  Competitive auctions and digital goods , 2001, SODA '01.

[28]  Ronitt Rubinfeld,et al.  Testing that distributions are close , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[29]  Frank Thomson Leighton,et al.  The value of knowing a demand curve: bounds on regret for online posted-price auctions , 2003, 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings..

[30]  Artur Czumaj,et al.  Abstract Combinatorial Programs and Efficient Property Testers , 2005, SIAM J. Comput..

[31]  Oded Goldreich.  Property testing in massive graphs , 2002 .

[32]  Nicole Immorlica,et al.  Multi-unit auctions with budget-constrained bidders , 2005, EC '05.

[33]  Dana Ron,et al.  Property Testing , 2000 .

[34]  Artur Czumaj,et al.  Testing Hereditary Properties of Nonexpanding Bounded-Degree Graphs , 2009, SIAM J. Comput..

[35]  D. Ron,et al.  Testing properties of directed graphs: acyclicity and connectivity , 2002 .

[36]  Noga Alon,et al.  lambda1, Isoperimetric inequalities for graphs, and superconcentrators , 1985, J. Comb. Theory, Ser. B.

[37]  Andrew V. Goldberg,et al.  Competitiveness via consensus , 2003, SODA '03.