Small-Error Cascaded Group Testing

Daniel McMorrow; Nikhil Karamchandani; Sidharth Jaggi

arxiv: 2601.11945 · v2 · submitted 2026-01-17 · 💻 cs.IT · math.IT

Small-Error Cascaded Group Testing

Daniel McMorrow , Nikhil Karamchandani , Sidharth Jaggi This is my paper

Pith reviewed 2026-05-16 13:18 UTC · model grok-4.3

classification 💻 cs.IT math.IT

keywords cascaded group testingsmall-error recoverynon-adaptive testingadaptive testingconstrained test sizelower boundsdefective set recovery

0 comments

The pith

Cascaded group testing with ordered tests that identify the first defective achieves small-error recovery, with matching lower bounds up to logs under constrained test sizes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that when each test returns the index of the earliest defective item in a pre-specified ordering, reliable recovery of the defective set becomes possible with fewer tests than standard binary group testing. Achievability bounds are derived for multiple recovery criteria under both non-adaptive and adaptive designs, and these hold whether test sizes are unconstrained or fixed in advance. In the constrained-size regime the upper bounds are shown to be tight up to logarithmic factors via an information-theoretic lower bound. A sympathetic reader would care because the extra information per test can materially reduce the total number of experiments needed in screening or fault-location tasks.

Core claim

In the cascaded group testing model, each test is supplied with an ordering of the items it contains and its outcome exactly identifies the first defective item according to that ordering. The authors prove that this model permits small-error recovery of the defective set via explicit constructions for both non-adaptive and adaptive testing, across several natural recovery criteria, and for both unconstrained and size-constrained tests. They further establish a matching lower bound in the constrained-test-size setting, proving that the number of tests required is optimal up to logarithmic factors.

What carries the argument

The cascaded test outcome that returns the precise index of the first defective item under a known ordering of the tested items.

If this is right

Non-adaptive designs suffice for small-error recovery at the stated rates.
Adaptive testing can be used without changing the asymptotic bounds.
Multiple recovery criteria (exact, approximate, etc.) are simultaneously achievable.
In the constrained-test-size case the upper bound is optimal up to logs.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The model may extend naturally to settings where the ordering can be chosen dynamically based on prior outcomes.
Applications in pooled screening could exploit controllable ordering to reduce total tests below binary-group-testing requirements.
The lower-bound technique may carry over to other non-binary group-testing variants that return partial location information.

Load-bearing premise

Each test must be supplied in advance with a fixed ordering of its items, and the outcome must return the exact index of the first defective item according to that ordering.

What would settle it

An explicit construction or information-theoretic argument showing that, under the same constrained test-size regime, the number of tests needed to achieve small-error recovery must grow faster than the paper's upper bound by more than a logarithmic factor.

read the original abstract

Group testing concerns itself with the accurate recovery of a set of "defective" items from a larger population via a series of tests. While most works in this area have considered the classical group testing model, where tests are binary and indicate the presence of at least one defective item in the test, we study the cascaded group testing model. In cascaded group testing, tests admit an ordering, and test outcomes indicate the first defective item in the test under this ordering. Under this model, we establish various achievability bounds for several different recovery criteria using both non-adaptive and adaptive test designs when assuming both unconstrained and constrained test sizes. In the constrained test size setting, we also provide a lower bound showing our achievability result is optimal up to logarithmic factors.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper introduces a cascaded group testing model with ordered tests that return the first defective index and gives concrete small-error achievability bounds plus a near-matching lower bound under test-size constraints.

read the letter

The main point is that they define a new variant where each test has a fixed ordering of items and the outcome tells you the index of the first defective one under that order. This is not the usual binary positive/negative test, and they derive small-error recovery bounds for both non-adaptive and adaptive designs, with and without limits on how many items you can put in a test. In the constrained-size case they also give a lower bound showing their achievability is optimal up to logarithmic factors. That combination of model and bounds is what is actually new here. The work does a clean job of stating the model and using standard mutual-information and counting arguments to get the results; nothing looks circular or self-referential. The logarithmic gap in the lower bound is normal for these combinatorial search problems and does not undermine the claim. The main soft spot is that the model requires the ordering to be known in advance and the test to reliably return the exact index, which narrows the practical settings where it applies. Small-error recovery is also weaker than exact identification, though they are explicit about the criterion. Without the full proofs it is hard to check the precise constants or dependence on the number of defectives, but the abstract and stress-test give no sign of internal contradictions. This is for people already working in combinatorial search theory or information-theoretic group testing. A reader in that subfield will find the new model and the constrained-size optimality result useful. It is grounded enough and the claims are specific enough that it deserves a serious referee rather than a desk reject.

Referee Report

0 major / 3 minor

Summary. The paper introduces the cascaded group testing model, in which each test is equipped with a known ordering of items and returns the index of the first defective item according to that ordering. It derives achievability bounds for small-error recovery of the defective set under non-adaptive and adaptive testing, both with unconstrained and constrained test sizes. In the constrained-test-size regime it also supplies a matching lower bound that is optimal up to logarithmic factors.

Significance. If the derivations hold, the work supplies a clean information-theoretic characterization of a group-testing variant whose test outcomes are strictly richer than the classical binary model. The near-tight lower bound in the constrained setting is a notable strength, as it places the achievable test complexity on a firm footing relative to standard combinatorial benchmarks. The results could inform designs in applications where positional information is obtainable at modest additional cost.

minor comments (3)

The abstract states achievability and lower bounds but does not display the explicit scaling with the number of defectives d or the target error probability; adding these scalings (even in big-O form) would make the central claims immediately verifiable from the abstract.
The model definition requires that the ordering be known in advance and that each test return a precise index; a short paragraph in the introduction clarifying how this assumption is realized in practice would help readers assess applicability.
Notation for the population size n, number of defectives d, and test-size constraint should be introduced once and used consistently; minor inconsistencies in the early sections could be eliminated by a uniform definition list.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our work on cascaded group testing, including the recognition of the near-tight lower bound under test-size constraints. The recommendation for minor revision is noted, and we will use the opportunity to improve presentation and clarity.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper establishes achievability bounds (non-adaptive and adaptive, unconstrained and constrained test sizes) plus a matching lower bound up to logarithmic factors for the explicitly defined cascaded group testing model, where each test returns the index of the first defective item under a known ordering. These results rest on standard combinatorial counting and information-theoretic arguments benchmarked against external combinatorial search problems, without any self-definitional reductions, fitted inputs renamed as predictions, load-bearing self-citations, imported uniqueness theorems, smuggled ansatzes, or renamings of known results. The model definition and optimality claim (accounting for the typical logarithmic gap) are self-contained and do not reduce to their own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. The model implicitly assumes a known ordering per test and exact identification of the minimal defective index, but these are definitional rather than fitted quantities.

pith-pipeline@v0.9.0 · 5427 in / 1080 out tokens · 39795 ms · 2026-05-16T13:18:10.511856+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We form the estimate ˆK by taking it to be the union of all test outcomes... leads to a coupon collector’s problem... P(ˆK ≠ K) ≤ k (1−1/k)^T
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean embed_injective unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

each test contains all items in N and the ordering ≤t is given by a uniformly random permutation of N

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages · 1 internal anchor

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work page 2014
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A. Coja-Oghlan, O. Gebhard, M. Hahn-Klimroth, and P. Loick, “Information-theoretic and algorithmic thresholds for group test- ing,”IEEE Transactions on Information Theory, vol. 66, no. 12, pp. 7911–7928, 2020

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M. Aldridge, O. Johnson, J. Scarlett,et al., “Group testing: an information theory perspective,”Foundations and Trends® in Com- munications and Information Theory, vol. 15, no. 3-4, pp. 196–392, 2019

work page 2019
[25]

Cascading bandits: Learning to rank in the cascade model,

B. Kveton, C. Szepesvari, Z. Wen, and A. Ashkan, “Cascading bandits: Learning to rank in the cascade model,” inInternational conference on machine learning, pp. 767–776, PMLR, 2015

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Combinatorial cascading bandits,

B. Kveton, Z. Wen, A. Ashkan, and C. Szepesvari, “Combinatorial cascading bandits,”Advances in Neural Information Processing Systems, vol. 28, 2015

work page 2015
[27]

Cost-aware cascading ban- dits,

C. Gan, R. Zhou, J. Yang, and C. Shen, “Cost-aware cascading ban- dits,”IEEE Transactions on Signal Processing, vol. 68, pp. 3692– 3706, 2020

work page 2020
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M. Falahatgar, A. Jafarpour, A. Orlitsky, V . Pichapati, and A. T. Suresh, “Estimating the number of defectives with group testing,” in 2016 IEEE International Symposium on Information Theory (ISIT), pp. 1376–1380, IEEE, 2016

work page 2016
[30]

Improved lower bound for estimating the number of defective items,

N. H. Bshouty, “Improved lower bound for estimating the number of defective items,”Journal of Combinatorial Optimization, vol. 49, no. 2, p. 33, 2025

work page 2025
[31]

The coupon collector’s prob- lem,

M. Ferrante and M. Saltalamacchia, “The coupon collector’s prob- lem,”Materials matematics, pp. 0001–35, 2014

work page 2014
[32]

The Probability Distribution for Draws Until First Success Without Replacement

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[33]

Mean estimation and regression under heavy-tailed distributions: A survey,

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The detection of defective members of large pop- ulations,

R. Dorfman, “The detection of defective members of large pop- ulations,”The Annals of mathematical statistics, vol. 14, no. 4, pp. 436–440, 1943

work page 1943

[2] [2]

A pooling strategy for heterozygote screening of theΔF508 cystic fibrosis mutation,

C. Gille, K. Grade, and C. Coutelle, “A pooling strategy for heterozygote screening of theΔF508 cystic fibrosis mutation,” Human Genetics, vol. 86, pp. 289–291, 1991

work page 1991

[3] [3]

Biological screens from linear codes: theory and tools,

Y . Erlich, A. Gilbert, H. Ngo, A. Rudra, N. Thierry-Mieg, M. Woot- ters, D. Zielinski, and O. Zuk, “Biological screens from linear codes: theory and tools,”BioRxiv, p. 035352, 2015

work page 2015

[4] [4]

Pooling DNA in the identification of parents,

R. N. Curnow and A. P. Morris, “Pooling DNA in the identification of parents,”Heredity, vol. 80, no. 1, pp. 101–109, 1998

work page 1998

[5] [5]

Born again group testing: Multiaccess communications,

J. Wolf, “Born again group testing: Multiaccess communications,” IEEE Transactions on Information Theory, vol. 31, no. 2, pp. 185– 191, 1985

work page 1985

[6] [6]

An asymptotically fast nonadaptive algorithm for conflict resolution in multiple-access channels,

J. Koml ´os and A. Greenberg, “An asymptotically fast nonadaptive algorithm for conflict resolution in multiple-access channels,”IEEE Transactions on Information Theory, vol. 31, no. 2, pp. 302–306, 1985

work page 1985

[7] [7]

Group testing for sensor networks: the value of asking the right questions,

Y .-W. Hong and A. Scaglione, “Group testing for sensor networks: the value of asking the right questions,” inConference Record of the Thirty-Eighth Asilomar Conference on Signals, Systems and Computers, vol. 2, pp. 1297–1301, IEEE, 2004

work page 2004

[8] [8]

Sparse group testing codes for low-energy massive random access,

H. A. Inan, P. Kairouz, and A. Ozgur, “Sparse group testing codes for low-energy massive random access,” in2017 55th Annual Allerton Conference on Communication, Control, and Computing (Allerton), pp. 658–665, IEEE, 2017

work page 2017

[9] [9]

What’s hot and what’s not: tracking most frequent items dynamically,

G. Cormode and S. Muthukrishnan, “What’s hot and what’s not: tracking most frequent items dynamically,”ACM Transactions on Database Systems (TODS), vol. 30, no. 1, pp. 249–278, 2005

work page 2005

[10] [10]

Cascaded group testing,

W. Mirza, N. Karamchandani, and N. Balachandran, “Cascaded group testing,” in2024 IEEE Information Theory Workshop (ITW), pp. 390–395, IEEE, 2024

work page 2024

[11] [11]

Bounds on the length of disjunctive codes,

A. G. D’yachkov and V . V . Rykov, “Bounds on the length of disjunctive codes,”Problemy Peredachi Informatsii, vol. 18, no. 3, pp. 7–13, 1982

work page 1982

[12] [12]

On r-cover-free families,

Z. F ¨uredi, “On r-cover-free families,”Journal of Combinatorial Theory, Series A, vol. 73, no. 1, pp. 172–173, 1996

work page 1996

[13] [13]

Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms,

C. L. Chan, P. H. Che, S. Jaggi, and V . Saligrama, “Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms,” in2011 49th Annual Allerton Conference on Communication, Control, and Computing (Allerton), pp. 1832–1839, IEEE, 2011

work page 2011

[14] [14]

The capacity of adaptive group testing,

L. Baldassini, O. Johnson, and M. Aldridge, “The capacity of adaptive group testing,” in2013 IEEE International Symposium on Information Theory, pp. 2676–2680, IEEE, 2013

work page 2013

[15] [15]

Optimal group testing,

A. Coja-Oghlan, O. Gebhard, M. Hahn-Klimroth, and P. Loick, “Optimal group testing,” inConference on Learning Theory, pp. 1374–1388, PMLR, 2020

work page 2020

[16] [16]

A method for detecting all defective members in a population by group testing,

F. K. Hwang, “A method for detecting all defective members in a population by group testing,”Journal of the American Statistical Association, vol. 67, no. 339, pp. 605–608, 1972

work page 1972

[17] [17]

Explicit non-adaptive combinatorial group testing schemes,

E. Porat and A. Rothschild, “Explicit non-adaptive combinatorial group testing schemes,” inInternational Colloquium on Automata, Languages, and Programming, pp. 748–759, Springer, 2008

work page 2008

[18] [18]

Individual testing is optimal for nonadaptive group testing in the linear regime,

M. Aldridge, “Individual testing is optimal for nonadaptive group testing in the linear regime,”IEEE Transactions on Information Theory, vol. 65, no. 4, pp. 2058–2061, 2018

work page 2058

[19] [19]

Optimal non-adaptive proba- bilistic group testing in general sparsity regimes,

W. H. Bay, J. Scarlett, and E. Price, “Optimal non-adaptive proba- bilistic group testing in general sparsity regimes,”Information and Inference: A Journal of the IMA, vol. 11, no. 3, pp. 1037–1053, 2022

work page 2022

[20] [20]

Non- adaptive group testing: Explicit bounds and novel algorithms,

C. L. Chan, S. Jaggi, V . Saligrama, and S. Agnihotri, “Non- adaptive group testing: Explicit bounds and novel algorithms,” IEEE Transactions on Information Theory, vol. 60, no. 5, pp. 3019– 3035, 2014

work page 2014

[21] [21]

Group testing algo- rithms: Bounds and simulations,

M. Aldridge, L. Baldassini, and O. Johnson, “Group testing algo- rithms: Bounds and simulations,”IEEE Transactions on Informa- tion Theory, vol. 60, no. 6, pp. 3671–3687, 2014

work page 2014

[22] [22]

Phase transitions in group testing,

J. Scarlett and V . Cevher, “Phase transitions in group testing,” in Proceedings of the twenty-seventh annual ACM-SIAM symposium on Discrete algorithms, pp. 40–53, SIAM, 2016

work page 2016

[23] [23]

Information-theoretic and algorithmic thresholds for group test- ing,

A. Coja-Oghlan, O. Gebhard, M. Hahn-Klimroth, and P. Loick, “Information-theoretic and algorithmic thresholds for group test- ing,”IEEE Transactions on Information Theory, vol. 66, no. 12, pp. 7911–7928, 2020

work page 2020

[24] [24]

Group testing: an information theory perspective,

M. Aldridge, O. Johnson, J. Scarlett,et al., “Group testing: an information theory perspective,”Foundations and Trends® in Com- munications and Information Theory, vol. 15, no. 3-4, pp. 196–392, 2019

work page 2019

[25] [25]

Cascading bandits: Learning to rank in the cascade model,

B. Kveton, C. Szepesvari, Z. Wen, and A. Ashkan, “Cascading bandits: Learning to rank in the cascade model,” inInternational conference on machine learning, pp. 767–776, PMLR, 2015

work page 2015

[26] [26]

Combinatorial cascading bandits,

B. Kveton, Z. Wen, A. Ashkan, and C. Szepesvari, “Combinatorial cascading bandits,”Advances in Neural Information Processing Systems, vol. 28, 2015

work page 2015

[27] [27]

Cost-aware cascading ban- dits,

C. Gan, R. Zhou, J. Yang, and C. Shen, “Cost-aware cascading ban- dits,”IEEE Transactions on Signal Processing, vol. 68, pp. 3692– 3706, 2020

work page 2020

[28] [28]

T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, Introduction to algorithms. MIT press, 2022

work page 2022

[29] [29]

Estimating the number of defectives with group testing,

M. Falahatgar, A. Jafarpour, A. Orlitsky, V . Pichapati, and A. T. Suresh, “Estimating the number of defectives with group testing,” in 2016 IEEE International Symposium on Information Theory (ISIT), pp. 1376–1380, IEEE, 2016

work page 2016

[30] [30]

Improved lower bound for estimating the number of defective items,

N. H. Bshouty, “Improved lower bound for estimating the number of defective items,”Journal of Combinatorial Optimization, vol. 49, no. 2, p. 33, 2025

work page 2025

[31] [31]

The coupon collector’s prob- lem,

M. Ferrante and M. Saltalamacchia, “The coupon collector’s prob- lem,”Materials matematics, pp. 0001–35, 2014

work page 2014

[32] [32]

The Probability Distribution for Draws Until First Success Without Replacement

J. Ahlgren, “The probability distribution for draws until first success without replacement,”arXiv preprint arXiv:1404.1161, 2014

work page internal anchor Pith review Pith/arXiv arXiv 2014

[33] [33]

Mean estimation and regression under heavy-tailed distributions: A survey,

G. Lugosi and S. Mendelson, “Mean estimation and regression under heavy-tailed distributions: A survey,”Foundations of Com- putational Mathematics, vol. 19, no. 5, pp. 1145–1190, 2019

work page 2019