Lower bounds for identifying subset members with subset queries
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The classical Group Testing Problem is: Given a finite set of items {1, 2, ... , n} and an unknown subset P⊆{1, 2, ..., n} of up to p positive elements, identify P by asking the least number of queries of the type “does the subset Q⊆{0, 2,..., n} intersect P?”. In our case, Q must be a subset of consecutive elements. This problem naturally arises in several scenarios, most notably in Computational Biology. We focus on algorithms in which queries are arranged in stages: in each stage, queries can be performed in parallel, and be chosen depending on the answers to queries in previous stages. Algorithms that operate in few stages are usually preferred in practice. First we study the case p = 1 comprehensively. For two-stage strategies for arbitrary p we obtain asymptotically tight bounds on the number of queries. Furthermore we prove bounds for any number of stages and positives, and we discuss the problem with the restriction that query intervals have some bounded length d.