Group Testing with Geometric Ranges

Group testing is a well-studied approach for identifying t defective items in a set X of m items, by testing appropriately chosen subsets of X. In classical group testing any subset of X can be tested, and for $t \in {\mathcal{O}}(1)$ the optimal number of (non-adaptive) tests is known to be Θ(logm).In this work we consider a novel geometric setting for group testing, where the items are points in Euclidean space and the tests are axis-parallel boxes (hyperrectangles), corresponding to the scenario where tests are defined by parameter-ranges (say, according to physical measurements). We present upper and lower bounds on the required number of tests in this setting, observing that in contrast to the unrestricted, combinatorial case, the bounds are polynomial in m. For instance, we show that with two parameters, identifying a defective pair of items requires Ω(m3/5) tests, and there exist configurations for which ${\mathcal{O}}\left({{m^{2/3}}}\right)$ tests are sufficient, whereas to identify a single defective item Θ(m1/2) tests are always necessary and sometimes sufficient. Perhaps most interestingly, our work brings to the study of group testing a set of techniques from extremal combinatorics.