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Testing Intersecting and Union-Closed Families

Authors Xi Chen, Anindya De, Yuhao Li, Shivam Nadimpalli, Rocco A. Servedio

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ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
  • Mathematics of computing → Combinatorics
Keywords
  • Sublinear algorithms
  • property testing
  • computational complexity
  • monotonicity
  • intersecting families
  • union-closed families

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Abstract

Inspired by the classic problem of Boolean function monotonicity testing, we investigate the testability of other well-studied properties of combinatorial finite set systems, specifically intersecting families and union-closed families. A function f: {0,1}ⁿ → {0,1} is intersecting (respectively, union-closed) if its set of satisfying assignments corresponds to an intersecting family (respectively, a union-closed family) of subsets of [n]. 
Our main results are that - in sharp contrast with the property of being a monotone set system - the property of being an intersecting set system, and the property of being a union-closed set system, both turn out to be information-theoretically difficult to test. We show that: 
- For ε ≥ Ω(1/√n), any non-adaptive two-sided ε-tester for intersectingness must make 2^{Ω(n^{1/4}/√{ε})} queries. We also give a 2^{Ω(√{n log(1/ε)})}-query lower bound for non-adaptive one-sided ε-testers for intersectingness.
- For ε ≥ 1/2^{Ω(n^{0.49})}, any non-adaptive two-sided ε-tester for union-closedness must make n^{Ω(log(1/ε))} queries.
Thus, neither intersectingness nor union-closedness shares the poly(n,1/ε)-query non-adaptive testability that is enjoyed by monotonicity.
To complement our lower bounds, we also give a simple poly(n^{√{nlog(1/ε)}},1/ε)-query, one-sided, non-adaptive algorithm for ε-testing each of these properties (intersectingness and union-closedness). We thus achieve nearly tight upper and lower bounds for two-sided testing of intersectingness when ε = Θ(1/√n), and for one-sided testing of intersectingness when ε = Θ(1).

Cite As Get BibTex

Xi Chen, Anindya De, Yuhao Li, Shivam Nadimpalli, and Rocco A. Servedio. Testing Intersecting and Union-Closed Families. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 33:1-33:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ITCS.2024.33

Author Details

Xi Chen
  • Columbia University, New York, NY, USA
Anindya De
  • University of Pennsylvania, Philadelphia, PA, USA
Yuhao Li
  • Columbia University, New York, NY, USA
Shivam Nadimpalli
  • Columbia University, New York, NY, USA
Rocco A. Servedio
  • Columbia University, New York, NY, USA

Funding

  • Chen, Xi: NSF grants IIS-1838154, CCF-2106429, and CCF-2107187.
  • De, Anindya: NSF grants CCF-1910534 and CCF-2045128.
  • Li, Yuhao: NSF grants IIS-1838154, CCF-2106429 and CCF-2107187.
  • Nadimpalli, Shivam: NSF grants IIS-1838154, CCF-2106429, CCF-2211238, CCF-1763970, and CCF-2107187.
  • Servedio, Rocco A.: NSF grants IIS-1838154, CCF-2106429, and CCF-2211238.

Acknowledgements

This work was partially completed while some of the authors were visiting the Simons Institute for the Theory of Computing at UC Berkeley.

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