A 1.431-Competitive Algorithm for Combinatorial Group Testing

In the context of fault-detection problems, the objective is to identify all defective items among a set of $n$ binary-state items using the minimum number of tests. The {group testing} paradigm, which allows testing a subset of items in a single test, serves as a fundamental technique for efficiently classifying large populations. We study a central problem in the combinatorial group testing model where the number $d$ of defective items is unknown in advance. Let $M_{\alpha }(d\mathrm{\mid }n)$ denote the maximum number of tests required by an algorithm $\alpha$ for this problem, and $M(d,n)$ denote the minimum number of tests required in the worst case when $d$ is known in advance. An algorithm $\alpha$ is called a $c$-\emph{competitive algorithm} if there exist constants $c$ and $a$ such that, for $0\le d<n$, $M_{\alpha }(d\mathrm{\mid }n)\le cM(d,n)+a$. We design a new adaptive algorithm with a competitive constant $c\le 1.431$, thus pushing the competitive ratio below the best-known one of $1.452$. To achieve this, we propose a novel solution framework based on an unexplored up-zig-zag strategy and a studied strongly competitive algorithm.