Unconstraining Graph-Constrained Group Testing

In network tomography, one goal is to identify a small set of failed links in a network using as little information as possible. One way of setting up this problem is called graph-constrained group testing. Graph-constrained group testing is a variant of the classical combinatorial group testing problem, where the tests that one is allowed are additionally constrained by a graph. In this case, the graph is given by the underlying network topology. The main contribution of this work is to show that for most graphs, the constraints imposed by the graph are no constraint at all. That is, the number of tests required to identify the failed links in graph-constrained group testing is near-optimal even for the corresponding group testing problem with no graph constraints. Our approach is based on a simple randomized construction of tests. To analyze our construction, we prove new results about the size of giant components in randomly sparsified graphs. Finally, we provide empirical results which suggest that our connected-subgraph tests perform better not just in theory but also in practice, and in particular perform better on a real-world network topology

Optimal Dorfman Group Testing For Symmetric Distributions

We study Dorfman's classical group testing protocol in a novel setting where individual specimen statuses are modeled as exchangeable random variables. We are motivated by infectious disease screening. In that case, specimens which arrive together for testing often originate from the same community and so their statuses may exhibit positive correlation. Dorfman's protocol screens a population of n specimens for a binary trait by partitioning it into nonoverlapping groups, testing these, and only individually retesting the specimens of each positive group. The partition is chosen to minimize the expected number of tests under a probabilistic model of specimen statuses. We relax the typical assumption that these are independent and indentically distributed and instead model them as exchangeable random variables. In this case, their joint distribution is symmetric in the sense that it is invariant under permutations. We give a characterization of such distributions in terms of a function q where q(h) is the marginal probability that any group of size h tests negative. We use this interpretable representation to show that the set partitioning problem arising in Dorfman's protocol can be reduced to an integer partitioning problem and efficiently solved. We apply these tools to an empirical dataset from the COVID-19 pandemic. The methodology helps explain the unexpectedly high empirical efficiency reported by the original investigators.Comment: 20 pages w/o references, 2 figure