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arxiv: 2606.27127 · v1 · pith:VFDCPU7Nnew · submitted 2026-06-25 · 💻 cs.IT · math.IT

Algorithms for Threshold Group Testing

Amin Coja-Oghlan , Remco van der Hofstad , Lena Krieg , Noela M\"uller , Connor Riddlesden , Olga Scheftelowitsch This is my paper

Pith reviewed 2026-06-26 02:09 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords threshold group testingnon-adaptive testingexact recoverysparse recoveryspatially coupled designsinformation-theoretic threshold
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The pith

An efficient algorithm recovers exact defectives in threshold group testing at the information-theoretic minimum number of non-adaptive tests.

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

The paper focuses on threshold group testing without noise or adaptivity, where a test on a pool returns positive only when the number of defectives meets or exceeds a fixed threshold. It presents an inference algorithm that uses a spatially coupled test design to achieve exact recovery of a sparse binary vector with high probability. The method operates at the smallest number of tests required by constant-column designs and thereby attains the known sharp information-theoretic threshold under an analytic condition. A reader would care because the algorithm avoids the weighted-sum analyses of earlier group-testing methods while still delivering near-optimal guarantees.

Core claim

The algorithm achieves exact recovery with high probability in the noiseless non-adaptive threshold group testing setting using the minimum number of tests needed for the constant-column design, thereby matching the information-theoretic threshold.

What carries the argument

Spatially coupled test design that structures the tests so the inference procedure admits a direct analysis without intricate weighted sums.

If this is right

  • Exact recovery is possible at the information-theoretic limit for this class of threshold group testing problems.
  • The algorithm applies directly to constant-column test designs.
  • The analysis remains simpler than the weighted-sum techniques used for binary group testing.
  • High-probability exact recovery holds at the minimum test count required by the design.

Where Pith is reading between the lines

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

  • The spatial-coupling construction may transfer to other sparse-recovery settings that exhibit similar phase transitions.
  • Implementations could exploit the simpler proof structure to obtain practical performance bounds more easily.
  • Relaxing the analytic condition would widen the range of threshold values covered by the guarantee.

Load-bearing premise

An analytic condition under which threshold group testing undergoes a sharp information-theoretic phase transition for exact recovery on constant-column designs.

What would settle it

An explicit constant-column test matrix and defective set where the algorithm fails to output the correct vector with high probability when the number of tests equals the information-theoretic minimum.

Figures

Figures reproduced from arXiv: 2606.27127 by Amin Coja-Oghlan, Connor Riddlesden, Lena Krieg, Noela M\"uller, Olga Scheftelowitsch, Remco van der Hofstad.

Figure 1
Figure 1. Figure 1: Graphical representation of a pooling scheme for [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The constant csc of visualised for t = 1, t = 2, t = 10. Theorem 1.1. For any 0 < θ < 1, ε > 0 and t ∈ N + fixed, there exists n0 = n0(θ, t, ε) such that, for every n > n0, there exist a randomised test design G with m ≤ (1 + ε)msc(n, θ, t) tests and a deterministic polynomial-time inference algorithm SPOT for which P [SPOT(G,σˆG) = σ] > 1 − ε. (4) [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Schematic representation of the pooling scheme with [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
read the original abstract

We study the Threshold Group Testing (TGT) problem without a gap in the noiseless, non-adaptive setting, where the goal is to exactly recover a sparse binary vector from pooled test outcomes using as few tests as possible. In TGT, a test applied to a subset of items returns a positive outcome if the number of defective items in the subset reaches a prescribed threshold, and a negative outcome otherwise. Under the assumption of an analytic condition, TGT has been shown to undergo a sharp information-theoretic phase transition for exact recovery on the class of constant-column test designs. In this paper, we develop an efficient inference algorithm that achieves exact recovery with high probability using the minimum number of non-adaptive tests that are needed for the constant-column design, thereby matching the information-theoretic threshold of a natural benchmark test design. Our approach is based on a spatially coupled test design and admits a significantly simpler analysis than existing algorithms for related group testing problems. In particular, unlike previous methods for binary group testing, our algorithm does not rely on the analysis of intricate weighted sums. This leads to a more straightforward proof technique, while still allowing near-optimal performance guarantees.

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.

Referee Report

1 major / 0 minor

Summary. The paper studies the noiseless non-adaptive Threshold Group Testing (TGT) problem. It develops an efficient inference algorithm based on a spatially coupled constant-column test design that achieves exact recovery with high probability at the information-theoretic minimum number of tests, matching the sharp phase transition established for constant-column designs under an assumed analytic condition. The analysis is presented as simpler than prior work because it avoids intricate weighted-sum arguments.

Significance. If the central claim holds, the result supplies a more accessible proof technique for near-optimal TGT recovery and demonstrates that spatially coupled constructions can attain the benchmark threshold without the analytic overhead of earlier methods. This could lower the barrier to extending such guarantees to related sparse-recovery settings.

major comments (1)
  1. [Abstract] Abstract: The claim that the algorithm matches the information-theoretic threshold rests on the sharp phase transition shown in prior work for constant-column designs under an unspecified analytic condition. The manuscript invokes this threshold directly for the spatially coupled construction but supplies no verification that the analytic condition continues to hold for the new design or in the operating regime of interest. This assumption is load-bearing for the matching claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting this important point about the information-theoretic threshold claim. We address the concern directly below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The claim that the algorithm matches the information-theoretic threshold rests on the sharp phase transition shown in prior work for constant-column designs under an unspecified analytic condition. The manuscript invokes this threshold directly for the spatially coupled construction but supplies no verification that the analytic condition continues to hold for the new design or in the operating regime of interest. This assumption is load-bearing for the matching claim.

    Authors: We agree that the analytic condition is central to invoking the sharp threshold from prior work on constant-column designs. Our spatially coupled construction is a constant-column design whose parameters are chosen to lie within the regime where the condition was previously verified; the spatial coupling preserves the necessary regularity (uniform column weight and bounded row degrees) that the condition depends on. Nevertheless, the manuscript does not explicitly confirm this for the coupled case. In the revision we will add a short paragraph (or appendix entry) that directly checks the analytic condition for the design parameters and operating regime used in our theorems. This will make the matching claim self-contained without changing any results or proofs. revision: yes

Circularity Check

0 steps flagged

No significant circularity; algorithm analysis is independent of cited benchmark

full rationale

The paper cites prior work establishing the information-theoretic threshold for constant-column designs under an analytic condition, then constructs a spatially coupled design and provides a new, simpler analysis showing that its algorithm achieves exact recovery at that same number of tests. The derivation chain for the algorithm's performance guarantee does not reduce to the prior threshold result by construction, self-definition, or fitted inputs; the benchmark serves only as an external performance target. No load-bearing self-citation chain or ansatz smuggling is exhibited in the abstract or described claims. The result is therefore self-contained against the external benchmark.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on an external analytic condition that produces the phase transition for constant-column designs; no free parameters, new axioms, or invented entities are introduced in the abstract itself.

axioms (1)
  • domain assumption An analytic condition produces a sharp information-theoretic phase transition for exact recovery on constant-column test designs.
    Invoked in the abstract as the foundation for the information-theoretic threshold that the algorithm matches.

pith-pipeline@v0.9.1-grok · 5746 in / 1248 out tokens · 15354 ms · 2026-06-26T02:09:36.051652+00:00 · methodology

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Reference graph

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