Learning Immune-Defectives Graph through Group Tests

This paper deals with an abstraction of a unified problem of drug discovery and pathogen identification. Here, the \"lead compounds\" are abstracted as inhibitors, pathogenic proteins as defectives, and the mixture of \"ineffective\" chemical compounds and non-pathogenic proteins as normal items. A defective could be immune to the presence of an inhibitor in a test. So, a test containing a defective is positive iff it does not contain its \"associated\" inhibitor. The goal of this paper is to identify the defectives, inhibitors, and their \"associations\" with high probability, or in other words, learn the Immune Defectives Graph (IDG). We propose a probabilistic non-adaptive pooling design, a probabilistic two-stage adaptive pooling design and decoding algorithms for learning the IDG. For the two-stage adaptive-pooling design, we show that the sample complexity of the number of tests required to guarantee recovery of the inhibitors, defectives and their associations with high probability, i.e., the upper bound, exceeds the proposed lower bound by a logarithmic multiplicative factor in the number of items. For the non-adaptive pooling design, in the large inhibitor regime, we show that the upper bound exceeds the proposed lower bound by a logarithmic multiplicative factor in the number of inhibitors.