Quickly-Decodable Group Testing with Fewer Tests: Price-Scarlett and Cheraghchi-Nakos's Nonadaptive Splitting with Explicit Scalars
<|reference_start|>Quickly-Decodable Group Testing with Fewer Tests: Price-Scarlett and Cheraghchi-Nakos's Nonadaptive Splitting with Explicit Scalars: We modify Cheraghchi-Nakos [CN20] and Price-Scarlett's [PS20] fast binary splitting approach to nonadaptive group testing. We show that, to identify a uniformly random subset of $k$ infected persons among a population of $n$, it takes only $\ln(2 - 4\varepsilon) ^{-2} k \ln n$ tests and decoding complexity $O(\varepsilon^{-2} k \ln n)$, for any small $\varepsilon > 0$, with vanishing error probability. In works prior to ours, only two types of group testing schemes exist. Those that use $\ln(2)^{-2} k \ln n$ or fewer tests require linear-in-$n$ complexity, sometimes even polynomial in $n$; those that enjoy sub-$n$ complexity employ $O(k \ln n)$ tests, where the big-$O$ scalar is implicit, presumably greater than $\ln(2)^{-2}$. We almost achieve the best of both worlds, namely, the almost-$\ln(2)^{-2}$ scalar and the sub-$n$ decoding complexity. How much further one can reduce the scalar $\ln(2)^{-2}$ remains an open problem.<|reference_end|>
@article{wang2024quickly-decodable,
    title={Quickly-Decodable Group Testing with Fewer Tests: Price-Scarlett and
  Cheraghchi-Nakos's Nonadaptive Splitting with Explicit Scalars},
    author={Hsin-Po Wang, Ryan Gabrys, Venkatesan Guruswami},
    journal={arXiv preprint arXiv:2405.16370},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.16370},
    primaryClass={cs.IT math.IT}
}