The goal of the group testing problem is to identify a set of defective items within a larger set of items, using suitably-designed tests whose outcomes indicate whether any defective item is present. In this paper, we study how the number of tests can be significantly decreased by leveraging the structural dependencies between the items, i.e., by incorporating prior information. To do so, we pursue two different perspectives: (i) As a generalization of the uniform combinatorial prior, we consider the case that the defective set is uniform over a \emph{subset} of all possible sets of a given size, and study how this impacts the information-theoretic limits on the number of tests for approximate recovery; (ii) As a generalization of the i.i.d.~prior, we introduce a new class of priors based on the Ising model, where the associated graph represents interactions between items. We show that this naturally leads to an Integer Quadratic Program decoder, which can be converted to an Integer Linear Program and/or relaxed to a non-integer variant for improved computational complexity, while maintaining strong empirical recovery performance.
翻译:组测试问题的目的是利用适当设计的测试结果显示是否存在任何有缺陷的项目。 在本文中,我们研究如何通过利用项目之间的结构依赖性,即纳入先前的信息,大幅降低测试数量。 为此,我们从两个不同的角度探讨:(一) 作为统一组合之前的概括,我们考虑一个案例,即缺陷集与所有可能大小的数组的 memph{subset 一致,并研究这如何影响信息理论限制对近似回收测试的数量的影响;(二) 作为i.i.d.~prior的概括,我们根据Ising 模型引入一个新的前期类别,其中相关的图表代表项目之间的相互作用。我们表明,这自然导致一个 Integer Quabaratic 程序解密器,可转换成 Integer 线条程序,并(或)向非 Interger 变异体,以改进计算复杂度,同时保持强有力的实证恢复性能。