CS: Graph-Constrained Group Testing

Found today on Arxiv. Interesting finding (emphasis below is mine):

Graph-Constrained Group Testing by Mahdi Cheraghchi, Amin Karbasi, Soheil Mohajer, Venkatesh Saligrama. The abstract reads:

Non-adaptive group testing involves grouping arbitrary subsets of $n$ items into different pools. Each pool is then tested and defective items are identified. A fundamental question involves minimizing the number of pools required to identify at most $d$ defective items. Motivated by applications in network tomography, sensor networks and infection propagation we formulate group testing problems on graphs. Unlike conventional group testing problems each group here must conform to the constraints imposed by a graph. For instance, items can be associated with vertices and each pool is any set of nodes that must be path connected. In this paper we associate a test with a random walk. In this context conventional group testing corresponds to the special case of a complete graph on $n$ vertices.
For interesting classes of graphs we arrive at a rather surprising result, namely, that the number of tests required to identify $d$ defective items is substantially similar to that required in conventional group testing problems, where no such constraints on pooling is imposed. Specifically, if T(n) corresponds to the mixing time of the graph $G$, we show that with $m=O(d^2 T^2(n) log(n/d))$ non-adaptive tests, one can identify the defective items. Consequently, for the Erdos-Renyi random graph $G(n,p)$, as well as expander graphs with constant spectral gap, it follows that $m=O(d^2 log^3(n))$ non-adaptive tests are sufficient to identify $d$ defective items. We next consider a specific scenario that arises in network tomography and show that $m=O(d^3 log^3(n))$ non-adaptive tests are sufficient to identify $d$ defective items. We also consider noisy counterparts of the graph constrained group testing problem and develop parallel results for these cases.

O(d^2 log^3(n)) uh !
Credit: SOHO, NASA and ESA