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Improved bounds for group testing in arbitrary hypergraphs

Annalisa De Bonis

TL;DR

This work develops a framework for group testing in arbitrary hypergraphs, where the contaminated set must be one of the hyperedges. It presents a sequence of few-stage algorithms, including a two-stage method achieving $o(d\log|E|)$ tests and a three-stage improvement by a factor $d^{1/6}$, plus a general $s$-stage construction with a new non-adaptive subroutine, enabling tunable adaptivity versus test count. It also derives a non-adaptive lower bound that approaches the best known upper bounds in certain regimes and provides a compression-based lower bound for $E$-separable codes. Collectively, these results advance the understanding of adaptivity-accuracy trade-offs in hypergraph-structured group testing and extend efficient testing strategies to contexts with social/geographical clustering encoded as hyperedges.

Abstract

Recent papers initiated the study of a generalization of group testing where the potentially contaminated sets are the members of a given hypergraph F=(V,E). This generalization finds application in contexts where contaminations can be conditioned by some kinds of social and geographical clusterings. The paper focuses on few-stage group testing algorithms, i.e., slightly adaptive algorithms where tests are performed in stages and all tests performed in the same stage should be decided at the very beginning of the stage. In particular, the paper presents the first two-stage algorithm that uses o(dlog|E|) tests for general hypergraphs with hyperedges of size at most d, and a three-stage algorithm that improves by a d^{1/6} factor on the number of tests of the best known three-stage algorithm. These algorithms are special cases of an s-stage algorithm designed for an arbitrary positive integer s<= d. The design of this algorithm resort to a new non-adaptive algorithm (one-stage algorithm), i.e., an algorithm where all tests must be decided beforehand. Further, we derive a lower bound for non-adaptive group testing. For E sufficiently large, the lower bound is very close to the upper bound on the number of tests of the best non-adaptive group testing algorithm known in the literature, and it is the first lower bound that improves on the information theoretic lower bound Omega(log |E|).

Improved bounds for group testing in arbitrary hypergraphs

TL;DR

This work develops a framework for group testing in arbitrary hypergraphs, where the contaminated set must be one of the hyperedges. It presents a sequence of few-stage algorithms, including a two-stage method achieving tests and a three-stage improvement by a factor , plus a general -stage construction with a new non-adaptive subroutine, enabling tunable adaptivity versus test count. It also derives a non-adaptive lower bound that approaches the best known upper bounds in certain regimes and provides a compression-based lower bound for -separable codes. Collectively, these results advance the understanding of adaptivity-accuracy trade-offs in hypergraph-structured group testing and extend efficient testing strategies to contexts with social/geographical clustering encoded as hyperedges.

Abstract

Recent papers initiated the study of a generalization of group testing where the potentially contaminated sets are the members of a given hypergraph F=(V,E). This generalization finds application in contexts where contaminations can be conditioned by some kinds of social and geographical clusterings. The paper focuses on few-stage group testing algorithms, i.e., slightly adaptive algorithms where tests are performed in stages and all tests performed in the same stage should be decided at the very beginning of the stage. In particular, the paper presents the first two-stage algorithm that uses o(dlog|E|) tests for general hypergraphs with hyperedges of size at most d, and a three-stage algorithm that improves by a d^{1/6} factor on the number of tests of the best known three-stage algorithm. These algorithms are special cases of an s-stage algorithm designed for an arbitrary positive integer s<= d. The design of this algorithm resort to a new non-adaptive algorithm (one-stage algorithm), i.e., an algorithm where all tests must be decided beforehand. Further, we derive a lower bound for non-adaptive group testing. For E sufficiently large, the lower bound is very close to the upper bound on the number of tests of the best non-adaptive group testing algorithm known in the literature, and it is the first lower bound that improves on the information theoretic lower bound Omega(log |E|).
Paper Structure (9 sections, 11 theorems, 14 equations)

This paper contains 9 sections, 11 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. Related work and our contribution
  3. Notations and terminology
  4. The group testing model
  5. Non-adaptive group testing for general hypergraphs
  6. Combinatorial structures for group testing in arbitrary hypergraphs
  7. Lower Bound on the Number of Rows of $E$-Separable Codes
  8. An improved non-adaptive group testing algorithm for general hypergraph
  9. An $s$-stage group testing algorithms for general hypergraphs

Key Result

lemma 1

Let $E$ be a set of hyperedges of size $d$ on $[n]$, and let $f$ be a positive integer with $f\leq |E|d/n$. The number of hyperedges of $|E|$ consisting only of vertices with degree larger than or equal to $f$ is at least $|E|-(f-1) n_s\geq |E|-(f-1) (n-1)$, where $n_s$ is the number of vertices in

Theorems & Definitions (16)

  • definition 1
  • lemma 1
  • proof
  • theorem 1
  • proof
  • corollary 1
  • corollary 2
  • theorem 2
  • theorem 3
  • proof
  • ...and 6 more