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Small error algorithms for tropical group testing

Vivekanand Paligadu, Oliver Johnson, Matthew Aldridge

TL;DR

The paper introduces tropical group testing, where test outcomes are the minimum defectivity level among pooled items, to capture Ct-like information from PCR testing. It analyzes non-adaptive tropical algorithms—COMP, DD, and SCOMP—through simulations and theoretical bounds, comparing them to classical binary group testing. A key result is that tropical COMP matches the classical counting bound asymptotically, while tropical DD (and SCOMP) achieve sharper thresholds and can be essentially optimal in denser regimes, highlighting a potential reduction in required tests by exploiting Ct-level information. Overall, the tropical model provides a tractable framework that leverages quantitative Ct data to improve pooling strategies without introducing test-loading delays, with implications for scalable PCR-like testing regimes.

Abstract

We consider a version of the classical group testing problem motivated by PCR testing for COVID-19. In the so-called tropical group testing model, the outcome of a test is the lowest cycle threshold (Ct) level of the individuals pooled within it, rather than a simple binary indicator variable. We introduce the tropical counterparts of three classical non-adaptive algorithms (COMP, DD and SCOMP), and analyse their behaviour through both simulations and bounds on error probabilities. By comparing the results of the tropical and classical algorithms, we gain insight into the extra information provided by learning the outcomes (Ct levels) of the tests. We show that in a limiting regime the tropical COMP algorithm requires as many tests as its classical counterpart, but that for sufficiently dense problems tropical DD can recover more information with fewer tests, and can be viewed as essentially optimal in certain regimes.

Small error algorithms for tropical group testing

TL;DR

The paper introduces tropical group testing, where test outcomes are the minimum defectivity level among pooled items, to capture Ct-like information from PCR testing. It analyzes non-adaptive tropical algorithms—COMP, DD, and SCOMP—through simulations and theoretical bounds, comparing them to classical binary group testing. A key result is that tropical COMP matches the classical counting bound asymptotically, while tropical DD (and SCOMP) achieve sharper thresholds and can be essentially optimal in denser regimes, highlighting a potential reduction in required tests by exploiting Ct-level information. Overall, the tropical model provides a tractable framework that leverages quantitative Ct data to improve pooling strategies without introducing test-loading delays, with implications for scalable PCR-like testing regimes.

Abstract

We consider a version of the classical group testing problem motivated by PCR testing for COVID-19. In the so-called tropical group testing model, the outcome of a test is the lowest cycle threshold (Ct) level of the individuals pooled within it, rather than a simple binary indicator variable. We introduce the tropical counterparts of three classical non-adaptive algorithms (COMP, DD and SCOMP), and analyse their behaviour through both simulations and bounds on error probabilities. By comparing the results of the tropical and classical algorithms, we gain insight into the extra information provided by learning the outcomes (Ct levels) of the tests. We show that in a limiting regime the tropical COMP algorithm requires as many tests as its classical counterpart, but that for sufficiently dense problems tropical DD can recover more information with fewer tests, and can be viewed as essentially optimal in certain regimes.
Paper Structure (12 sections, 5 theorems, 17 equations, 1 figure, 1 table, 3 algorithms)

This paper contains 12 sections, 5 theorems, 17 equations, 1 figure, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Group testing and tropical group testing
  3. The tropical model as an approximation of PCR testing
  4. Our contributions
  5. Notation and tropical model
  6. Description of tropical algorithms
  7. General remarks concerning algorithms
  8. Counting bounds
  9. Tropical COMP
  10. Tropical DD
  11. Tropical SCOMP
  12. Comparison of tropical algorithms

Key Result

Lemma 2.3

For each item $i$, we have $U_i \geq \mu_i$. In particular, if $\mu_i = \infty$ (that is, if the item appears in a negative test) then we can recover with certainty that $U_i = \infty$.

Figures (1)

  • Figure 1: Schematic illustration of test matrix and outcomes sorted into block form. Here a $0$ represents a submatrix of all zeroes, a $+1$ represents a submatrix which has at least one entry equal to $1$ in each column, and $?$ represents a submatrix which could be of any form. The defective items are sorted by level to the left of the vertical line. The column labels above the matrix represents the number of elements of each type; the vector represents the outcomes of the test.

Theorems & Definitions (17)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Definition 3.1
  • Theorem 3.2
  • proof
  • Definition 3.3
  • Lemma 3.4
  • ...and 7 more