One-Sided Group Testing

• One-sided group testing is a family of pooled-test models that incorporate asymmetry in hypotheses, noise, recovery, or cost, enhancing detection efficiency.
• It uses strategies such as weight-threshold decision rules, dynamic programming in adaptive screening, and combinatorial constructions to balance error types.
• Applications range from defective-count hypothesis testing and dilution noise models to selective error recovery and positive-response minimization.

Searching arXiv for the specified topic and related group-testing literature. A one-sided group test is a pooled-testing formulation in which the asymmetry of the problem is explicit in the hypotheses, the response channel, or the admissible reconstruction errors. In the classical Boolean model, a test on a subset of items is negative if all items in the subset are non-defective and positive if at least one defective item is present. Within that baseline, one-sided formulations include composite tests of whether the number of defectives is at most $s_1$ or at least $s_2$, noise models with false negatives but no false positives, recovery criteria that permit only false negatives or only false positives, and objectives that minimize the number of positive responses rather than only the total number of tests (D'yachkov et al., 2017, Arpino et al., 2021, McMorrow et al., 12 Jun 2025, Bonis, 2016). This suggests that “one-sided group test” is not a single canonical model but a family of asymmetric group-testing problems built around the same pooled-observation paradigm.

1. Terminological scope and core formalism

The common substrate is a population of $n$ items and a defective subset $S\subseteq\{1,\dots,n\}$. In the standard non-adaptive OR model, the test-design matrix $\mathbf X\in\{0,1\}^{T\times n}$ specifies which items are included in each pool, and the $t$-th outcome is

$Y_t=\bigvee_{i\in S}X_{t,i}.$

In adaptive screening formulations, the same binary semantics are used, but subsequent tests may depend on prior outcomes. In probabilistic screening models, items are defective independently with probabilities $p$ or $p_i$, and the objective is to minimize the expected number of tests needed to identify all statuses (Malinovsky et al., 2021).

The asymmetry that makes a formulation “one-sided” varies across papers. In one line of work, the asymmetry is inferential: decide $H_0:D\le s_1$ versus $s_2$0, where $s_2$1 and the ambiguous range $s_2$2 is excluded (D'yachkov et al., 2017). In another, the asymmetry is in the error model: dilution noise flips $s_2$3 with probability $s_2$4, so the channel has false negatives only (Arpino et al., 2021). In a third, the asymmetry is in the reconstruction target: approximate recovery is constrained to allow only false negatives or only false positives (McMorrow et al., 12 Jun 2025). In yet another, the asymmetry is operational: positive tests are treated as costly, and the problem is to minimize the number of “yes” responses subject to a budget of total tests (Bonis, 2016).

Formulation	One-sided feature	Representative result
Defective-count hypothesis test	$s_2$5 vs. $s_2$6	$s_2$7-WDR (D'yachkov et al., 2017)
Dilution-noise model	false negatives only	$s_2$8 and $s_2$9 scaling (Arpino et al., 2021)
Approximate recovery	only FN or only FP allowed	SUBSET and SUPERSET criteria (McMorrow et al., 12 Jun 2025)
Positive-response minimization	minimize “yes” responses	adaptive, non-adaptive, and two-stage bounds (Bonis, 2016)
Generalized responses	positivity depends on density or max load	density-based and max-load tests (Gerbner et al., 2012, Joly et al., 2020)

A common misconception is that one-sidedness refers only to the Boolean outcome rule itself. The literature indicates otherwise: the same phrase is used for asymmetry in hypotheses, channel noise, decoding guarantees, and test costs.

2. One-sided hypothesis testing on the number of defectives

A particularly direct formulation appears in "Hypothesis Test for Bounds on the Size of Random Defective Set" (D'yachkov et al., 2017). The problem is to decide between the composite hypotheses

$n$0

with integers $n$1. The design is an $n$2 Bernoulli matrix $n$3 with i.i.d. entries $n$4, and each test outcome $n$5 is Boolean.

The proposed decoder is the weight-threshold decision rule, denoted $n$6-WDR. It computes the number of positive tests

$n$7

and compares $n$8 with a fixed threshold $n$9: if $S\subseteq\{1,\dots,n\}$0, declare $S\subseteq\{1,\dots,n\}$1; if $S\subseteq\{1,\dots,n\}$2, declare $S\subseteq\{1,\dots,n\}$3. Under the boundary cases $S\subseteq\{1,\dots,n\}$4 and $S\subseteq\{1,\dots,n\}$5, the single-test positive probabilities are

$S\subseteq\{1,\dots,n\}$6

To balance type I and type II errors, the threshold is chosen as

$S\subseteq\{1,\dots,n\}$7

The error analysis is based on Hoeffding’s inequality. Writing $S\subseteq\{1,\dots,n\}$8, the type I and type II errors satisfy

$S\subseteq\{1,\dots,n\}$9

A near-optimal design parameter is

$\mathbf X\in\{0,1\}^{T\times n}$0

With this choice, the sample size sufficient for error at most $\mathbf X\in\{0,1\}^{T\times n}$1 is

$\mathbf X\in\{0,1\}^{T\times n}$2

The asymptotic behavior depends on the gap between $\mathbf X\in\{0,1\}^{T\times n}$3 and $\mathbf X\in\{0,1\}^{T\times n}$4. If $\mathbf X\in\{0,1\}^{T\times n}$5 with fixed $\mathbf X\in\{0,1\}^{T\times n}$6, then $\mathbf X\in\{0,1\}^{T\times n}$7, independent of $\mathbf X\in\{0,1\}^{T\times n}$8. If $\mathbf X\in\{0,1\}^{T\times n}$9 with $t$0, then $t$1 (D'yachkov et al., 2017). This is the key distinction from full defective-set recovery: when only the composite boundary matters, the number of tests need not scale with the population size. The same source contrasts this with COMP, whose test complexity is $t$2, and reports computational complexity $t$3 for the weight-threshold rule versus $t$4 for COMP.

The paper also identifies a worst-case structure for the composite uncertainty: the maximal error of $t$5-WDR over all prior distributions on $t$6 is attained by any two-mass prior with $t$7 (D'yachkov et al., 2017). This reduces the composite problem to the two boundary cases for analysis, which is typical of one-sided composite tests.

3. One-sided recovery guarantees and one-sided noise

A distinct use of one-sidedness concerns which reconstruction errors are allowed. In "Optimal Non-Adaptive Group Testing with One-Sided Error Guarantees" (McMorrow et al., 12 Jun 2025), the noiseless OR model is retained, but exact recovery is relaxed in two asymmetric ways. Under SUBSET recovery, the goal is $t$8 with $t$9, so only false negatives are allowed. Under SUPERSET recovery, the goal is $Y_t=\bigvee_{i\in S}X_{t,i}.$0 with $Y_t=\bigvee_{i\in S}X_{t,i}.$1, so only false positives are allowed. In the sparse regime $Y_t=\bigvee_{i\in S}X_{t,i}.$2, the rate is defined by

$Y_t=\bigvee_{i\in S}X_{t,i}.$3

For SUBSET recovery, Theorem 2 shows that rate $Y_t=\bigvee_{i\in S}X_{t,i}.$4 is achievable, equivalently

$Y_t=\bigvee_{i\in S}X_{t,i}.$5

while ensuring $Y_t=\bigvee_{i\in S}X_{t,i}.$6 and $Y_t=\bigvee_{i\in S}X_{t,i}.$7, provided

$Y_t=\bigvee_{i\in S}X_{t,i}.$8

for any fixed $Y_t=\bigvee_{i\in S}X_{t,i}.$9. The construction uses Bernoulli testing with

$p$0

followed by a two-step decoder: first a two-sided approximate-recovery decoder with distortion $p$1, then an exhaustive search over subsets $p$2 of size $p$3 near the preliminary estimate, returning the candidate that explains the most tests (McMorrow et al., 12 Jun 2025). The paper states that this second step is exponential in general.

For SUPERSET recovery, Theorem 3 gives a converse: no non-adaptive scheme can exceed rate

$p$4

when $p$5. Achievability in the very sparse regime is obtained by COMP with near-constant-column-weight design: each item appears in $p$6 tests, any item appearing in a negative test is marked non-defective, and all remaining items are declared defective. Because defectives never appear in negative tests, the output automatically satisfies $p$7 (McMorrow et al., 12 Jun 2025). The asymmetry between SUBSET and SUPERSET is substantive rather than linguistic: allowing only false negatives can preserve the counting-bound rate $p$8, whereas allowing only false positives leads to the threshold $p$9.

One-sidedness also arises in the observation channel. "Group Testing in the High Dilution Regime" studies a dilution model in which each included defective is independently diluted with probability $p_i$0, so its contribution is erased; the test is positive if and only if at least one undegraded defective participates (Arpino et al., 2021). The design is Bernoulli with

$p_i$1

which increases the inclusion probability as the dilution level rises. The decoder is $p_i$2: for each item $p_i$3, let $p_i$4 be the number of tests containing $p_i$5 and $p_i$6 the number of positive tests containing $p_i$7; declare $p_i$8 defective if

$p_i$9

For $H_0:D\le s_1$0, the paper proves an achievability bound

$H_0:D\le s_1$1

and an algorithm-independent converse

$H_0:D\le s_1$2

so the bounds match up to order in the high-noise regime (Arpino et al., 2021). The notable phenomenon is that dilution noise can be offset by choosing a suitable noise-level-dependent Bernoulli design.

4. Adaptive screening and nested one-sided procedures

In screening applications, the phrase one-sided group testing is often used for the classical binary pooled test itself: negative means every item in the group is non-defective, and positive means at least one item is defective. "Nested Group Testing Procedures for Screening" reviews this model under independent Bernoulli defectivity, with either homogeneous probability $H_0:D\le s_1$3 or heterogeneous probabilities $H_0:D\le s_1$4, and studies adaptive procedures that completely identify all item statuses with minimum expected number of tests (Malinovsky et al., 2021).

The basic two-stage scheme is Dorfman’s procedure. If pools have size $H_0:D\le s_1$5, then one pooled test is applied to each group; a negative result clears all $H_0:D\le s_1$6 items, while a positive result triggers individual testing of the group. The expected number of tests per person is

$H_0:D\le s_1$7

For small $H_0:D\le s_1$8, the optimal pool size satisfies $H_0:D\le s_1$9 and the minimal per-person cost satisfies $s_2$00 (Malinovsky et al., 2021). The modified Dorfman procedure $s_2$01 removes the unnecessary final individual test when the last person’s status is implied.

Sterrett’s sequential procedure $s_2$02 exploits the first positive individual found in a positive pool and recursively applies $s_2$03 to the remainder. Its expected tests per person are

$s_2$04

For $s_2$05, the optimal Sterrett pool size satisfies $s_2$06 and

$s_2$07

(Malinovsky et al., 2021). Thus Sterrett asymptotically halves Dorfman’s small-$s_2$08 cost.

More general nested strategies are obtained by dynamic programming. The hierarchical procedure $s_2$09 is optimal within hierarchical algorithms and has an $s_2$10 dynamic program. The fully nested procedure $s_2$11, which distinguishes between a “binomial set” and a “defective set” known a priori to contain at least one defective, has a dynamic program of $s_2$12 in the formulation summarized in the review (Malinovsky et al., 2021). The same paper records a Shannon-entropy bound: $s_2$13 and, for $s_2$14, reports the ranking $s_2$15 across the tabulated values of $s_2$16.

The significance of this screening literature is that it fixes the binary one-sided test as the primitive measurement and shifts the optimization to adaptive decision trees, expected cost, and heterogeneity in prior defect probabilities.

5. Generalized one-sided response models

The OR response is not the only one-sided pooling rule studied in the literature. "Density-based group testing" replaces the rule “positive iff at least one defective is present” by a threshold condition: $s_2$17 for a fixed $s_2$18 (Gerbner et al., 2012). The objective is to identify at least $s_2$19 defectives when $s_2$20. With $s_2$21, the main adaptive strategy tests disjoint blocks of size $s_2$22 until enough positive blocks are found, then performs binary-search-style extraction inside each positive block. When the current universe has defective density at least $s_2$23, halving preserves the density condition and isolates a defective in $s_2$24 further queries.

For fixed $s_2$25 and $s_2$26, Theorem 7 gives

$s_2$27

hence

$s_2$28

for large $s_2$29 (Gerbner et al., 2012). This model is one-sided because positivity requires sufficiently many defectives relative to pool size; the usual OR model is recovered only at the extreme threshold behavior.

A different generalization appears in "A tractable non-adaptative group testing method for non-binary measurements" (Joly et al., 2020). Here each defective item carries a positive load $s_2$30, non-defectives have load $s_2$31, and a test returns

$s_2$32

The design is a deterministic multipool construction on an $s_2$33 grid with rows, columns, and diagonals, under which there are exactly $s_2$34 pools, each pool has size $s_2$35, each item lies in exactly $s_2$36 pools, and any two items co-occur in at most one pool. Hence the total number of tests is $s_2$37, and the efficiency is $s_2$38.

Decoding is local. For each item $s_2$39, let $s_2$40 be the pools containing it and define

$s_2$41

If $s_2$42, declare the item negative. Otherwise count the number of pools in $s_2$43 attaining the minimum; if that count is at least $s_2$44, declare the item positive with estimated load $s_2$45 (Joly et al., 2020). The paper derives explicit bounds on false negatives and false positives via

$s_2$46

and, in the infinite-precision regime $s_2$47, shows that the optimal efficiency satisfies

$s_2$48

A plausible implication is that one-sided group testing can exploit richer measurement alphabets while retaining structurally simple decoding rules.

6. Positive-response minimization and combinatorial constructions

A further asymmetry arises when positive tests are themselves the scarce resource. "Constraining the Number of Positive Responses in Adaptive, Non-Adaptive, and Two-Stage Group Testing" considers a universe $s_2$49 of size $s_2$50 containing at most $s_2$51 defectives, a budget of at most $s_2$52 total tests, and the quantity $s_2$53, the number of positive responses (Bonis, 2016). For an algorithm $s_2$54, $s_2$55 is the maximum number of positives incurred over all defective sets of size at most $s_2$56, and

$s_2$57

is the optimal value.

The combinatorial objects governing the non-adaptive and two-stage cases are $s_2$58-cover-free and $s_2$59-cover-free families. A family $s_2$60 is $s_2$61-cover-free if no member is contained in the union of any other $s_2$62 members; equivalently, its characteristic-vector code is $s_2$63-disjunct. It is $s_2$64-cover-free if for every choice of $s_2$65 members $s_2$66 and any other $s_2$67 members $s_2$68, there is an element in $s_2$69 that does not lie in $s_2$70 (Bonis, 2016).

For adaptive testing, the paper derives a lower bound on the number of positives and an adaptive algorithm that matches the information-theoretic lower bound up to an additive $s_2$71. The algorithm is a multi-stage halving scheme: partition the current candidate set into groups, test the groups in parallel, keep only groups testing positive, and iterate until the defectives are isolated (Bonis, 2016). For non-adaptive testing, the paper gives almost matching upper and lower bounds on $s_2$72, including both probabilistic and explicit constructions. One explicit construction uses linear codes and yields a deterministic non-adaptive scheme with

$s_2$73

positive tests.

For two-stage testing, the main qualitative result is that the positive-response budget asymptotically matches the adaptive setting: $s_2$74 (Bonis, 2016). The paper describes this as an analogue of the classical phenomenon that “two stages suffice.” In this formulation, one-sidedness is not about the observation channel or the error criterion, but about a cost function concentrated on positive outcomes.

Across these lines of work, one-sided group testing is best understood as a structured asymmetry imposed on pooled inference. The asymmetry may concern the null and alternative classes, the sign of admissible errors, the noise channel, the positivity rule, or the cost assigned to positive tests. The resulting models differ substantially in design, decoding, and achievable rates, but they are unified by a common principle: once the asymmetry is specified, optimality hinges on aligning the pooling design and the decision rule with that single favored direction.