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Testing Coverage Functions

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Part of the Lecture Notes in Computer Science book series (LNTCS,volume 7391)

Abstract

A coverage function f over a ground set [m] is associated with a universe U of weighted elements and m sets A 1,…,A m  ⊆ U, and for any T ⊆ [m], f(T) is defined as the total weight of the elements in the union ∪  j ∈ T A j . Coverage functions are an important special case of submodular functions, and arise in many applications, for instance as a class of utility functions of agents in combinatorial auctions.

Set functions such as coverage functions often lack succinct representations, and in algorithmic applications, an access to a value oracle is assumed. In this paper, we ask whether one can test if a given oracle is that of a coverage function or not. We demonstrate an algorithm which makes O(m|U|) queries to an oracle of a coverage function and completely reconstructs it. This gives a polytime tester for succinct coverage functions for which |U| is polynomially bounded in m. In contrast, we demonstrate a set function which is “far” from coverage, but requires \(2^{\tilde{\Theta}(m)}\) queries to distinguish it from the class of coverage functions.

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References

  1. Badanidiyuru, A., Dobzinski, S., Fu, H., Kleinberg, R., Nisan, N., Roughgarden, T.: Sketching valuation functions. In: SODA (2012)

    Google Scholar 

  2. Bertsimas, D., Tsitsiklis, J.: Introduction to linear optimization. Athena Scientific Belmont, MA (1997)

    Google Scholar 

  3. Blumrosen, L., Nisan, N.: Combinatorial auctions. Algorithmic Game Theory (2007)

    Google Scholar 

  4. Chakrabarty, D., Huang, Z.: Testing coverage functions. Arxiv (2012)

    Google Scholar 

  5. Cornuejols, G., Fisher, M., Nemhauser, G.: Location of bank accounts to optimize float: An analytic study of exact and approximate algorithms. Management Science, 789–810 (1977)

    Google Scholar 

  6. Dughmi, S., Roughgarden, T., Yan, Q.: From convex optimization to randomized mechanisms: toward optimal combinatorial auctions. In: STOC, pp. 149–158. ACM (2011)

    Google Scholar 

  7. Dughmi, S., Vondrák, J.: Limitations of randomized mechanisms for combinatorial auctions. In: FOCS (2011)

    Google Scholar 

  8. Goldreich, O.: Combinatorial property testing (a survey). Randomization Methods in Algorithm Design 43, 45–59 (1999)

    MathSciNet  Google Scholar 

  9. Goldreich, O., Goldwasser, S., Ron, D.: Property testing and its connection to learning and approximation. Journal of the ACM (JACM) 45(4), 653–750 (1998)

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. Krause, A., McMahan, H., Guestrin, C., Gupta, A.: Robust submodular observation selection. Journal of Machine Learning Research 9, 2761–2801 (2008)

    MATH  Google Scholar 

  11. Lehmann, B., Lehmann, D., Nisan, N.: Combinatorial auctions with decreasing marginal utilities. Games and Economic Behavior 55(2), 270–296 (2006)

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. Mahler, K.: Introduction to p-adic numbers and their functions (1973)

    Google Scholar 

  13. O’Donnell, R.: Chapter 3.5 highlight: The Goldreich-Levin algorithm. In: Analysis of Boolean Functions (2012), http://analysisofbooleanfunctions.org/

  14. Robert, A.: A course in p-adic analysis, vol. 198. Springer (2000)

    Google Scholar 

  15. Seshadhri, C.: Open problems 2: Open problems in data streams, property testing, and related topics (2011), http://www.cs.umass.edu/~mcgregor/papers/11-openproblems.pdf

  16. Seshadhri, C., Vondrák, J.: Is submodularity testable? In: ICS (2011)

    Google Scholar 

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Chakrabarty, D., Huang, Z. (2012). Testing Coverage Functions. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds) Automata, Languages, and Programming. ICALP 2012. Lecture Notes in Computer Science, vol 7391. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31594-7_15

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  • DOI: https://doi.org/10.1007/978-3-642-31594-7_15

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-31593-0

  • Online ISBN: 978-3-642-31594-7

  • eBook Packages: Computer ScienceComputer Science (R0)