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The goal of combinatorial group testing is to efficiently identify up to $d$defective items in a large population of $n$ items, where $d \ll n$. Defectiveitems satisfy certain properties while the remaining items in the population donot. To efficiently identify defective items, a subset of items is pooled andthen tested. In this work, we consider complex group testing (CmplxGT) in whicha set of defective items consists of subsets of positive items (called\textit{positive complexes}). CmplxGT is classified into two categories:classical CmplxGT (CCmplxGT) and generalized CmplxGT (GCmplxGT). In CCmplxGT,the outcome of a test on a subset of items is positive if the subset containsat least one positive complex, and negative otherwise. In GCmplxGT, the outcomeof a test on a subset of items is positive if the subset has a certain numberof items of some positive complex, and negative otherwise.  For CCmplxGT, we present a scheme that efficiently identifies all positivecomplexes in time $t \times \mathrm{poly}(d, \ln{n})$ in the presence oferroneous outcomes, where $t$ is a predefined parameter. As $d \ll n$, this issignificantly better than the currently best time of $\mathrm{poly}(t) \timesO(n \ln{n})$. Moreover, in specific cases, the number of tests in our proposedscheme is smaller than previous work. For GCmplxGT, we present a scheme thatefficiently identifies all positive complexes. These schemes are directlyapplicable in various areas such as complex disease genetics, molecularbiology, and learning a hidden graph.

Efficient and error-tolerant schemes for non-adaptive complex group testing and its application in complex disease genetics