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Testing Sumsets Is Hard

Authors Xi Chen , Shivam Nadimpalli , Tim Randolph , Rocco A. Servedio , Or Zamir

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  • Theory of computation → Streaming, sublinear and near linear time algorithms
Keywords
  • Sumsets
  • additive combinatorics
  • property testing
  • Boolean functions

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Abstract

A subset S of the Boolean hypercube 𝔽₂ⁿ is a sumset if S = {a + b : a, b ∈ A} for some A ⊆ 𝔽₂ⁿ. Sumsets are central objects of study in additive combinatorics, where they play a role in several of the field’s most important results. We prove a lower bound of Ω(2^{n/2}) for the number of queries needed to test whether a Boolean function f:𝔽₂ⁿ → {0,1} is the indicator function of a sumset, ruling out an efficient testing algorithm for sumsets.
Our lower bound for testing sumsets follows from sharp bounds on the related problem of shift testing, which may be of independent interest. We also give a near-optimal {2^{n/2} ⋅ poly(n)}-query algorithm for a smoothed analysis formulation of the sumset refutation problem. Finally, we include a simple proof that the number of different sumsets in 𝔽₂ⁿ is 2^{(1±o(1))2^{n-1}}.

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Xi Chen, Shivam Nadimpalli, Tim Randolph, Rocco A. Servedio, and Or Zamir. Testing Sumsets Is Hard. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 14:1-14:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ESA.2025.14

Author Details

Xi Chen
  • Department of Computer Science, Columbia University, New York, NY, USA
Shivam Nadimpalli
  • Department of Mathematics, MIT, Cambridge, MA, USA
Tim Randolph
  • Department of Computer Science, Harvey Mudd College, Claremont, CA, USA
Rocco A. Servedio
  • Department of Computer Science, Columbia University, New York, NY, USA
Or Zamir
  • Blavatnik School of Computer Science, Tel Aviv University, Israel

Funding

  • Chen, Xi: Supported in part by NSF awards CCF-2106429 and CCF-2107187.
  • Nadimpalli, Shivam: Supported in part by NSF grants CCF-2106429, CCF-2211238, CCF-1763970, and CCF-210718.
  • Servedio, Rocco A.: Supported in part by NSF awards CCF-2211238 and CCF-2106429.
  • Zamir, Or: Supported in part by the Israel Science Foundation, Grant No. 1593/24, and in part by the Blavatnik Family Foundation.

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