Testing Sumsets is Hard
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913222830850048 |
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| author | Chen, Xi Nadimpalli, Shivam Randolph, Tim Servedio, Rocco A. Zamir, Or |
| author_facet | Chen, Xi Nadimpalli, Shivam Randolph, Tim Servedio, Rocco A. Zamir, Or |
| contents | A subset $S$ of the Boolean hypercube $\mathbb{F}_2^n$ is a sumset if $S = \{a + b : a, b\in A\}$ for some $A \subseteq \mathbb{F}_2^n$. Sumsets are central objects of study in additive combinatorics, featuring in several influential results. We prove a lower bound of $Ω(2^{n/2})$ for the number of queries needed to test whether a Boolean function $f:\mathbb{F}_2^n \to \{0,1\}$ is the indicator function of a sumset. 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 $2^{n/2} \cdot \mathrm{poly}(n)$-query algorithm for a smoothed analysis formulation of the sumset refutation problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_07242 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Testing Sumsets is Hard Chen, Xi Nadimpalli, Shivam Randolph, Tim Servedio, Rocco A. Zamir, Or Data Structures and Algorithms Computational Complexity Combinatorics A subset $S$ of the Boolean hypercube $\mathbb{F}_2^n$ is a sumset if $S = \{a + b : a, b\in A\}$ for some $A \subseteq \mathbb{F}_2^n$. Sumsets are central objects of study in additive combinatorics, featuring in several influential results. We prove a lower bound of $Ω(2^{n/2})$ for the number of queries needed to test whether a Boolean function $f:\mathbb{F}_2^n \to \{0,1\}$ is the indicator function of a sumset. 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 $2^{n/2} \cdot \mathrm{poly}(n)$-query algorithm for a smoothed analysis formulation of the sumset refutation problem. |
| title | Testing Sumsets is Hard |
| topic | Data Structures and Algorithms Computational Complexity Combinatorics |
| url | https://arxiv.org/abs/2401.07242 |