Quickly-Decodable Group Testing with Fewer Tests: Price-Scarlett and Cheraghchi-Nakos's Nonadaptive Splitting with Explicit Scalars
Hsin-Po Wang, Ryan Gabrys, Venkatesan Guruswami
TL;DR
This work advances nonadaptive group testing by merging the sub-n decoding benefits of Price–Scarlett and Cheraghchi–Nakos with a refined analysis that pushes the test count toward COMP-like efficiency. The authors implement PCNS’s two-phase framework (grow-and-prune, then leaf-trimming) and introduce explicit scalar tuning via ck to closely match COMP’s code rate, while preserving sub-n decoding complexity. They further adapt the construction to obtain a DD-style variant with a slightly different test budget and a different complexity profile, both with vanishing error probability. The contributions are complemented by detailed WA and TLE analyses, yielding practical, fast decoding in sparse regimes and clarifying the trade-offs between test count, decoding complexity, and error modes in nonadaptive GT. The approach relies on Mobius-type transformations and generating function arguments to control the evolution of the suspect set and the total work, with explicit constants and high-probability guarantees.
Abstract
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.