Competitive Group Testing and Learning Hidden Vertex Covers with Minimum Adaptivity

Abstract

Suppose that we are given a set of n elements d of which are “defective”. A group test can check for any subset, called a pool, whether it contains a defective. It is well known that d defectives can be found by using O(dlogn) pools. This nearly optimal number of pools can be achieved in 2 stages, where tests within a stage are done in parallel. But then d must be known in advance. Here we explore group testing strategies that use a nearly optimal number of pools and a few stages although d is not known to the searcher. One easily sees that O(logd) stages are sufficient for a strategy with O(dlogn) pools. Here we prove a lower bound of ${\it \Omega}(\log d/\log\log d)$ stages and a more general pools vs. stages tradeoff. As opposed to this, we devise a randomized strategy that finds d defectives using O(dlog(n/d)) pools in 3 stages, with any desired probability 1 − ε. Open questions concern the optimal constant factors and practical implications. A related problem motivated by, e.g., biological network analysis is to learn hidden vertex covers of a small size k in unknown graphs by edge group tests. (Does a given subset of vertices contain an edge?) We give a 1-stage strategy using O(k ³logn) pools, with any FPT algorithm for vertex cover enumeration as a decoder.