Combinatorial Quantitative Group Testing with Adversarially Perturbed Measurements: Fundamental Limits and Algorithm Design

In this paper, combinatorial quantitative group testing (QGT) with noisy measurements is studied. The goal of QGT is to detect defective items from a data set of size $n$ with counting measurements, each of which counts the number of defects in a selected pool of items. While most literatures consider either probabilistic QGT with random noise or combinatorial QGT with noiseless measurements, our focus is on the combinatorial QGT with noisy measurements that might be adversarially perturbed by additive bounded noises. Since perfect detection is impossible, a partial detection criterion is adopted. With the adversarial noise being bounded by $d_n = \Theta(n^\delta)$ and the detection criterion being to ensure no more than $k_n = \Theta(n^\kappa)$ errors can be made, our goal is to characterize the fundamental limit on the number of measurement, termed \emph{pooling complexity}, as well as provide explicit construction of measurement plans with optimal pooling complexity and efficient decoding algorithms. We first show that the fundamental limit is $\frac{1}{1-2\delta}\frac{n}{\log n}$ to within a constant factor not depending on $(n,\kappa,\delta)$ for the non-adaptive setting when $0<2\delta\leq \kappa <1$, sharpening the previous result by Chen and Wang [2]. We also provide an explicit construction of a non-adaptive deterministic measurement plan with $\frac{1}{1-2\delta}\frac{n}{\log_{2} n}$ pooling complexity up to a constant factor, matching the fundamental limit, with decoding complexity being $o(n^{1+\rho})$ for all $\rho > 0$, nearly linear in $n$, the size of the data set.