Efficient Designs for Threshold Group Testing Without Gap

Given $d$ defective items in a population of $n$ items with $d\ll n$, in threshold group testing without gap, the outcome of a test on a subset of items is positive if the subset has at least $u$ defective items and negative otherwise, where $1 \leq u \leq d$. The basic goal of threshold group testing is to quickly identify the defective items via a small number of tests. In non-adaptive design, all tests are designed indepen-dently and can be performed in parallel. The decoding time in the non-adaptive state-of-the-art work is a polynomial of $(d/u)^{u}(d/(d-u))^{d-u}, d$, and $\log n$. In this work, we present a novel design that significantly reduces the number of tests and the decoding time to polynomials of $\min\{u^{u},\ (d-u)^{d-u}\}, d$, and $\log n$. In particular, when $u$ is a constant, the number of tests and the decoding time are $O(d^{3}(\log^{2}n)\log(n/d))$ and $O(d^{3}(\log^{A}n)\log(n/d)+d^{2}(\log n)\log^{3}(n/d))$, respectively. For a special case when $u=2$, with non-adaptive design, the number of tests and the decoding time are $O(d^{3}(\log n)\log(n/d))$ and $O(d^{2}(\text{log} n+\log^{2}(n/d)))$, respectively. Moreover, with 2-stage design, the number of tests and the decoding time are $O(d^{2}\log^{2}(n/d))$. The full version is available at [1].