Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Concomitant Group Testing

Thach V. Bui, Jonathan Scarlett

TL;DR

Concomitant Group Testing (ConcGT) introduces a structured GT model where a test is positive only if it contains at least one item from each of m disjoint semi-defective sets. The paper develops a spectrum of deterministic and randomized designs across non-adaptive, adaptive, and limited-adaptivity regimes, achieving order-optimal or near-optimal test counts in many regimes, notably for small m. It leverages disjunct matrices for deterministic non-adaptive schemes, and constructs efficient randomized encodings and decodings that combine standard GT subroutines with novel subset-intersection strategies. The results substantially improve over general hypergraph-learning baselines, providing practical strategies for identifying multi-type defect interactions with significantly fewer tests, while outlining open questions for broader $m$ and non-adaptive least-resource scenarios.

Abstract

In this paper, we introduce a variation of the group testing problem capturing the idea that a positive test requires a combination of multiple ``types'' of item. Specifically, we assume that there are multiple disjoint \emph{semi-defective sets}, and a test is positive if and only if it contains at least one item from each of these sets. The goal is to reliably identify all of the semi-defective sets using as few tests as possible, and we refer to this problem as \textit{Concomitant Group Testing} (ConcGT). We derive a variety of algorithms for this task, focusing primarily on the case that there are two semi-defective sets. Our algorithms are distinguished by (i) whether they are deterministic (zero-error) or randomized (small-error), and (ii) whether they are non-adaptive, fully adaptive, or have limited adaptivity (e.g., 2 or 3 stages). Both our deterministic adaptive algorithm and our randomized algorithms (non-adaptive or limited adaptivity) are order-optimal in broad scaling regimes of interest, and improve significantly over baseline results that are based on solving a more general problem as an intermediate step (e.g., hypergraph learning).

Concomitant Group Testing

TL;DR

Concomitant Group Testing (ConcGT) introduces a structured GT model where a test is positive only if it contains at least one item from each of m disjoint semi-defective sets. The paper develops a spectrum of deterministic and randomized designs across non-adaptive, adaptive, and limited-adaptivity regimes, achieving order-optimal or near-optimal test counts in many regimes, notably for small m. It leverages disjunct matrices for deterministic non-adaptive schemes, and constructs efficient randomized encodings and decodings that combine standard GT subroutines with novel subset-intersection strategies. The results substantially improve over general hypergraph-learning baselines, providing practical strategies for identifying multi-type defect interactions with significantly fewer tests, while outlining open questions for broader and non-adaptive least-resource scenarios.

Abstract

In this paper, we introduce a variation of the group testing problem capturing the idea that a positive test requires a combination of multiple ``types'' of item. Specifically, we assume that there are multiple disjoint \emph{semi-defective sets}, and a test is positive if and only if it contains at least one item from each of these sets. The goal is to reliably identify all of the semi-defective sets using as few tests as possible, and we refer to this problem as \textit{Concomitant Group Testing} (ConcGT). We derive a variety of algorithms for this task, focusing primarily on the case that there are two semi-defective sets. Our algorithms are distinguished by (i) whether they are deterministic (zero-error) or randomized (small-error), and (ii) whether they are non-adaptive, fully adaptive, or have limited adaptivity (e.g., 2 or 3 stages). Both our deterministic adaptive algorithm and our randomized algorithms (non-adaptive or limited adaptivity) are order-optimal in broad scaling regimes of interest, and improve significantly over baseline results that are based on solving a more general problem as an intermediate step (e.g., hypergraph learning).
Paper Structure (32 sections, 5 theorems, 22 equations, 3 figures, 2 tables, 3 algorithms)

This paper contains 32 sections, 5 theorems, 22 equations, 3 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Motivation
  3. Problem formulation
  4. Related work
  5. Connection to complex GT and learning a hidden hypergraph
  6. Contributions
  7. Comparisons
  8. Overview of Techniques
  9. Deterministic Designs
  10. Non-adaptive design
  11. Correctness
  12. Complexity
  13. Adaptive design
  14. The case $m = 2$
  15. The case $m \geq 2$
  16. ...and 17 more sections

Key Result

Theorem 1

For the ConcGT problem with $m \ge 2$ disjoint nonempty subsets $S_1, \ldots, S_m \subset [n]$ with $|S_i| \leq s_i$, to deterministically recover $S_1, \ldots, S_m$ with a fixed number of tests (possibly adaptive), the required number of tests is at least Moreover, the same order-wise lower bound remains true even when the algorithm is given the following two kinds of flexibility: (i) it is rand

Figures (3)

  • Figure 1: Encoding procedure for the non-adaptive and randomized design. There are multiple phases that produce three types of vectors: indicator vectors, reference vectors, and identification vectors. The rows of $\mathcal{A}$ and $\mathcal{B}$ are generated using uniform sampling without replacement.
  • Figure 2: Decoding procedure for the non-adaptive and randomized design.
  • Figure 3: Encoding and decoding procedure for the two-stage and randomized design.

Theorems & Definitions (6)

  • Theorem 1
  • proof
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Lemma 1