Tighten the noisy-recovery test-complexity bound

Derive a tighter sufficient test-complexity bound for exact recovery by the graph total variation regularized decoder in the noisy non-adaptive group testing model, improving the current scaling from O(K log n) toward the noiseless O(K log(n/K)) behavior.

Background

The paper studies non-adaptive group testing when infections form localized clusters on a known contact graph. It proposes a graph total variation regularized decoder and proves sufficient conditions for exact recovery under a Bernoulli pooling design in both noiseless and noisy settings.

For the noisy setting, the stated sufficient condition requires a number of tests that scales as K log n for fixed auxiliary parameters. The authors note that this is worse than the K log(n/K) scaling obtained in the noiseless analysis and attribute the gap primarily to relaxations used in the proof. They explicitly leave the derivation of a tighter bound unresolved.

References

For fixed $\beta$, $\lambda$ and $\delta$, the number of tests required scales as $K\log n$. Compared with the noiseless case, the gap in scaling is primarily due to relaxations used in the proof. Deriving a tighter bound remains an open problem.

— Graph-Aware Group Testing with Locally Clustered Infections  (2609.20418 - Li et al., 17 Sep 2026) in Remark following Theorem "Sufficient Condition for Noisy Recovery," Section 4.3.2, "Sufficient Condition for Exact Recovery in the Noisy Setting"