Generalized Non-adaptive Group Testing
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In the problem of classical group testing one aims to identify a small subset (of expected size d) diseased individuals/defective items in a large population (of size n) based on a minimal number of suitably-designed group non-adaptive tests on subsets of items, where the test outcome is governed by an "OR" function, i.e., the test outcome is positive iff the given test contains at least one defective item. Motivated by physical considerations we consider a generalized scenario (that includes as special cases multiple other group-testing-like models in the literature) wherein the test outcome is governed by an arbitrary monotone (stochastic) test function f(·), with the test outcome being positive with probability f(x), where x is the number of defectives tested in that pool. For any monotone test function f(·) we present a non-adaptive generalized group-testing scheme that identifies all defective items with high probability. Our scheme requires at most O(d^2log(n)) tests for any monotone test function f(·), and at most O(dlog(n)) in the physically relevant sub-class of sensitive test functions (and hence is information-theoretically order-optimal for this sub-class).
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