A 1.431-Competitive Algorithm for Combinatorial Group Testing

TL;DR

This work addresses combinatorial group testing with an unknown number of defectives by introducing Z^c, a deterministic adaptive algorithm that couples an up-zig-zag strategy with a strongly competitive component. The authors prove a finite-sample competitive bound $M_{Z^c}(d|n) \le 1.431\,M(d,n)+39$, improving the previous best of $1.452$. The core technical contributions are two upper bounds for the up-zig-zag procedure $Z^u$ and a symmetric partitioning framework that orchestrates when to switch between subroutines based on preliminary testing outcomes. This yields a practically implementable strategy with a competitive ratio below 1.5, with potential for further refinements and extensions to adaptive defect estimation and tighter bounds.

Abstract

In the context of fault-detection problems, the objective is to identify all defective items among a set of $n$ binary-state items using the minimum number of tests. The {group testing} paradigm, which allows testing a subset of items in a single test, serves as a fundamental technique for efficiently classifying large populations. We study a central problem in the combinatorial group testing model where the number $d$ of defective items is unknown in advance. Let $M_α(d|n)$ denote the maximum number of tests required by an algorithm $α$ for this problem, and $M(d,n)$ denote the minimum number of tests required in the worst case when $d$ is known in advance. An algorithm $α$ is called a $c$-\emph{competitive algorithm} if there exist constants $c$ and $a$ such that, for $0\le d < n$, $M_α(d|n)\le cM(d,n)+a$. We design a new adaptive algorithm with a competitive constant $c \le 1.431$, thus pushing the competitive ratio below the best-known one of $1.452$. To achieve this, we propose a novel solution framework based on an unexplored up-zig-zag strategy and a studied strongly competitive algorithm.

Figures (3)

• Figure 1: . A Summary of Known Results for the Combinatorial Group Testing Model on $c$-Competitive and Strongly Competitive Algorithms Note. Between the two $x$-axes, the upper one (solid line) corresponds to $c$-competitive algorithms, while the lower one (dotted line) corresponds to strongly competitive algorithms.
• Figure 2: . An Example to Illustrate the Partition of $\mathscr{T}$ into Four Classes. Note.(a) The circles represent tests performed in Steps (5) and (13) (i.e., tests in $\mathscr{T}$), in order that they are performed by Procedure \ref{['alg:Zu']}. In this example, we have $\mathscr{T} = \{T_1, T_2,\cdots, T_{16}\}$. This test process contains $5$ phases, such as $\mathscr{P}_1 = \{T_1, T_2, T_3\}$, $\mathscr{P}_2 = \{T_4\}$, $\mathscr{P}_3 = \{T_5\}$, $\mathscr{P}_4 = \{T_6, T_7, T_8, T_9, T_{10}, T_{11}, T_{12}, T_{13}, T_{14}\}$ and $\mathscr{P}_5 = \{T_{15}, T_{16}\}$. An arrow pointing to the bottom right from a circle (test) $T$ indicates that the test result of $T$ is contaminated, and the rank of the test (performed in Step (13)) immediately following $T$ is decreased by one. An arrow pointing to top right from a test $T$ indicates that the test result of $T$ is pure, and if $T$ is the sixth test in this phase, the test (performed in Step (5)) immediately following $T$ is for the whole remaining items; otherwise, the rank of the test (performed in Step (13)) immediately following $T$ is increased by one. An arrow pointing to the top left from a test $T$ indicates that the test result of $T$ is contaminated and $T$ is an additional test. Assume the preceding test of an additional test is $T'$, then the rank of the test (performed in Step (13)) immediately following an additional test is increased by one based on $r(T')$, i.e., $r(T') + 1$. Notice that a test $T$ of rank zero may have a contaminated test result. In this case, a horizontal arrow is drawn from $T$ to the right, indicating that the next test in Step (13) also has rank zero. For this example, $C_1 = \{T_{15}, T_{16}\}$ with $\Delta_{C_1}=1$, and $C_2=\{T_5\}$. (b) Tests in $C_3$. In this example, $C_3$ is formed by three zig-zag tuples, and one possibility for the tuples could be $(T_1, T_4,\emptyset)$, $(T_7, T_3,\emptyset)$ and $(T_{13}, T_{14}, T_{12})$. Here $T_{12}$ is an additional test with $D(T_{12}) = D(T_{13})\cup D(T_{14})\cup D(T_{15})\cup D(T_{16})$. (c) Tests in $C_4 = \mathscr{T}\setminus (C_1\cup C_2\cup C_3) = \{T_6, T_2, T_8, T_9, T_{10}, T_{11}\}$. Notice that $T_2$ (gray circles) is a pure test for a subset {0,1} and $T_7$ is a pure test for a subset $\{0,0\}$ based on Remark \ref{['rem:redefine']}. So $C_3$ is not unique, for example, we can also have $C_3 = \{(T_1, T_4,\emptyset), (T_2, T_3,\emptyset),(T_{13}, T_{14}, T_{12})\}$.
• Figure 3: . An Example to Illustrate the Five Types of Tuples in Claim \ref{['ob:specialtuple']} Note. The serial numbers {(i),$\cdots$,(v)} in the figure correspond to the serial numbers in Claim \ref{['ob:specialtuple']}. In order to distinguish the different tests on $T^1$ and $T^2$, we use the percentage of black areas in a hollow circle to mark the number of defective items detected by a test. For example, $D(^2T^2) = \{0,1,1\}$ and so $^2T^2$ is a cycle covered by two-thirds of the black area.

Theorems & Definitions (28)

• Theorem 1
• Lemma 1
• Lemma 2
• Corollary 1
• Corollary 2
• Lemma 3
• Lemma 4
• Lemma 5
• Lemma 6
• Theorem 2