In many fault-detection problems, its objective lies in discerning all the defective items from a set of $n$ items with a binary characteristic by utilizing the minimum number of tests. Group testing (i.e., testing a subset of items with a single test) plays a pivotal role in classifying a large population of items. We study a central problem in the combinatorial group testing model for the situation where the number $d$ of defectives is unknown in advance. Let $M(d,n)=\min_{\alpha}M_\alpha(d|n)$, where $M_\alpha(d|n)$ is the maximum number of tests required by an algorithm $\alpha$ for the problem. An algorithm $\alpha$ is called a $c$\emph{-competitive algorithm} if there exist constants $c$ and $a$ such that for $0\le d < n$, $M_{\alpha}(d|n)\le cM(d,n)+a$. We design a new adaptive algorithm with competitive constant $c \le 1.431$, thus pushing the competitive ratio below the best-known one of $1.452$. The new algorithm is a novel solution framework based on a strongly competitive algorithm and an up-zig-zag strategy that has not yet been investigated.

翻译：在许多故障检测问题中，目标在于利用最少的测试次数，从一组具有二元特征的$n$个物品中识别出所有缺陷物品。分组测试（即单次测试一个物品子集）在对大量物品进行分类时起着关键作用。我们研究组合分组测试模型中的一个核心问题，即缺陷物品数量$d$事先未知的情况。设$M(d,n)=\min_{\alpha}M_\alpha(d|n)$，其中$M_\alpha(d|n)$是算法$\alpha$针对该问题所需的最大测试次数。若存在常数$c$和$a$，使得对于$0\le d < n$，有$M_{\alpha}(d|n)\le cM(d,n)+a$，则称算法$\alpha$为$c$-\emph{竞争算法}。我们设计了一种新的自适应算法，其竞争常数$c \le 1.431$，从而将竞争比推至低于已知最佳值1.452。该算法基于一种强竞争算法和一种尚未被研究的向上锯齿形策略，构成了一种新颖的解决方案框架。