Alapfogalmak
This paper presents improved bounds for few-stage group testing algorithms in arbitrary hypergraphs, where the potentially contaminated sets are the members of a given hypergraph.
Kivonat
The paper focuses on few-stage group testing algorithms, where tests are performed in stages and all tests in the same stage are decided at the beginning of the stage. The key contributions are:
The first two-stage algorithm that uses o(d log |E|) tests for general hypergraphs with hyperedges of size at most d.
A three-stage algorithm that improves by a d^(1/6) factor on the number of tests of the best known three-stage algorithm.
An s-stage algorithm designed for an arbitrary positive integer s ≤ d, where the number of tests decreases as s increases, providing a trade-off between the number of tests and adaptiveness.
The design of the s-stage algorithm relies on a new non-adaptive algorithm that uses O(b/p log |E|) tests to discard all hyperedges e that contain at least p non-defective vertices, provided that the size of the difference e' \ e between any two hyperedges e, e' is at most b.
A lower bound on the minimum length of non-adaptive group testing algorithms (E-separable codes) for the case when all hyperedges have size d. This lower bound is the first to improve on the information theoretic lower bound Ω(log |E|) and gets close to the best known upper bound.