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Anonymity-Preserving Space Partitions

Authors Úrsula Hébert-Johnson, Chinmay Sonar, Subhash Suri, Vaishali Surianarayanan

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Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Anonymity
  • Hitting Set
  • LP
  • Constant Approximation
  • Fixed-Parameter Tractable
  • Space Partitions
  • Parameterized Complexity

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Abstract

We consider a multidimensional space partitioning problem, which we call Anonymity-Preserving Partition. Given a set P of n points in ℝ^d and a collection H of m axis-parallel hyperplanes, the hyperplanes of H partition the space into an arrangement A(H) of rectangular cells. Given an integer parameter t > 0, we call a cell C in this arrangement deficient if 0 < |C ∩ P| < t; that is, the cell contains at least one but fewer than t data points of P. Our problem is to remove the minimum number of hyperplanes from H so that there are no deficient cells. We show that the problem is NP-complete for all dimensions d ≥ 2. We present a polynomial-time d-approximation algorithm, for any fixed d, and we also show that the problem can be solved exactly in time (2d-0.924)^k m^O(1) + O(n), where k is the solution size. The one-dimensional case of the problem, where all hyperplanes are parallel, can be solved optimally in polynomial time, but we show that a related Interval Anonymity problem is NP-complete even in one dimension.

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Úrsula Hébert-Johnson, Chinmay Sonar, Subhash Suri, and Vaishali Surianarayanan. Anonymity-Preserving Space Partitions. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 32:1-32:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.ISAAC.2021.32

Author Details

Úrsula Hébert-Johnson
  • Department of Computer Science, University of California, Santa Barbara, CA, USA
Chinmay Sonar
  • Department of Computer Science, University of California, Santa Barbara, CA, USA
Subhash Suri
  • Department of Computer Science, University of California, Santa Barbara, CA, USA
Vaishali Surianarayanan
  • Department of Computer Science, University of California, Santa Barbara, CA, USA

Funding

  • Suri, Subhash: in part supported by NSF grant CCF-1814172.

Acknowledgements

We thank Daniel Lokshtanov for discussions relating to our approximation algorithm for Anonymity-Preserving Partition, as well as an anonymous reviewer, whose comments helped to improve the aforementioned result.

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