Efficient Group Testing Algorithms for Arbitrary Hypergraphs

Belangrijkste concepten

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.

Samenvatting

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.

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Belangrijkste Inzichten Gedestilleerd Uit

Improved bounds for group testing in arbitrary hypergraphs

by Annalisa De ... om arxiv.org 04-30-2024

https://arxiv.org/pdf/2404.18783.pdf

Diepere vragen

How can the proposed group testing algorithms be extended or adapted to handle dynamic changes in the hypergraph structure, such as addition or removal of hyperedges

To adapt the proposed group testing algorithms for dynamic changes in the hypergraph structure, such as addition or removal of hyperedges, a few modifications can be made. One approach is to incorporate a mechanism to update the algorithm dynamically as the hypergraph changes. This can involve reevaluating the set of hyperedges, recalculating the tests needed based on the new structure, and adjusting the algorithm accordingly. By implementing a dynamic updating mechanism, the algorithm can effectively handle changes in the hypergraph structure without compromising its efficiency or accuracy.

What are the potential limitations or drawbacks of the assumption that the size of the difference between any two hyperedges is bounded by a constant b

While the assumption that the size of the difference between any two hyperedges is bounded by a constant b simplifies the algorithm design and analysis, it may introduce limitations in certain scenarios. One potential drawback is that this assumption may not hold in real-world hypergraphs, where the differences between hyperedges can vary significantly. If the actual differences between hyperedges exceed the specified constant b, the algorithm may not perform optimally and could potentially miss detecting the defective hyperedge. Therefore, the applicability of the algorithm may be limited to hypergraphs where the assumption of bounded differences is reasonable.

Can the techniques used in the design of the s-stage algorithm be applied to other combinatorial search problems beyond group testing

The techniques used in the design of the s-stage algorithm for group testing can be applied to other combinatorial search problems beyond group testing. The concept of breaking down the search process into multiple stages, each focusing on specific criteria or constraints, can be generalized to various optimization and search problems. By structuring the algorithm into stages that progressively narrow down the search space based on different parameters, it becomes possible to efficiently solve complex combinatorial problems. This approach can be adapted to problems in diverse fields such as network optimization, resource allocation, scheduling, and more, where breaking down the problem into manageable stages can lead to improved efficiency and effectiveness in finding solutions.