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 have a property called defective. A group test can check for any subset, called a pool, whether it contains a defective. It is known that a nearly optimal number of O(d log (n/d)) pools in two stages (where tests within a stage are done in parallel) are sufficient, but then the searcher must know d 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 beforehand. We prove a lower bound of Ω( log d/ log log d) stages and more general pools versus stages tradeoff. This is almost tight, since O( log d) stages are sufficient for a strategy with O(d log n) pools. As opposed to this negative result, we devise a randomized strategy using O(d log (n/d)) pools in three stages, with any desired success probability 1-ϵ. With some additional measures even two stages are enough. Open questions concern the optimal constant factors and practical implications. A related problem motivated by 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 one-stage strategy using O(k3 log n) pools, with any parameterized algorithm for vertex cover enumeration as a decoder. During the course of this work we also provide a classification of types of randomized search strategies in general.