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CS-432 · computer science · mathematics · Computational geometry

Fixed-Parameter Tractability of Zonotope Containment

Status Open · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
Standard number systems are denoted by $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$; index ranges, computational models and specialized notation follow the cited problem source and the definitions below.

The centered zonotope generated by $G$ is $Z(G)=\{Gx:x\in[-1,1]^n\}$. Containment means every point of the first zonotope belongs to the second. Dimension-parameterized runtime has a computable factor depending only on $d$.

Fixed-parameter tractability for parameter $k$ means running time $f(k)N^{O(1)}$, where $N$ is the input length and $f$ is computable. The exponent is independent of $k$.

Statement recorded in the supplied source.
Given $G\in\mathbb R^{d\times n}$ and $H\in\mathbb R^{d\times m}$, can one decide whether the zonotope $Z(G)$ is contained in $Z(H)$ in time $f(d)\operatorname{poly}(n+m)$ for some computable function $f$? [S2]
Scope qualification.
The source's centered/noncentered zonotope convention and real-input computational model are not specified in the short statement.
Source status.
The supplied catalogue labels this record open. Its source-status metadata was checked against the supplied snapshot on 2026-10-11. This import does not independently establish current open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-22.

Short English statement

Decide zonotope containment with dimension confined to the parameter-dependent runtime factor.

Sources

Preparation and review.
Original question attributed to the cited source and UnsolvedMath contributors. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, source inspection and formatting. No new human mathematical authorship or independent proof verification is asserted. Canonical ID 30006105 and source code OWR-14298804-007 checked against the complete supplied registry and existing repository statements; known flagged repetitions and obvious identical conjectures excluded. Full mathematical-equivalence/literature review has not been performed. Statement preparation and final review followed contribution prompts C and D; automated checks verify structure and identity, not open status.

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 2

CS-432-A2 · Exploration · LLM-generated · gpt 6 astra ultra ·

gpt 6 astra ultra attempt on CS-432

Duplicate Dimension-Parameterized Zonotope Containment

Definitions and mathematical statement

Shared notation.
Standard number systems are $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$. Specialized notation and index ranges follow the cited source and the definitions below.

The centered zonotope generated by $G$ is $Z(G)=\{Gx:x\in[-1,1]^n\}$. Containment means every point of the first zonotope belongs to the second. Dimension-parameterized runtime has a computable factor depending only on $d$.

Record scope and qualifications.
The catalogue title is retained as metadata. Named certificates, contracts, bounds or frameworks in that title are not asserted as results of the original source extract. The supplied registry flags this extraction as unrecoverable. It is displayed as a literal source fragment; the definitions provide context rather than a replacement formal statement.

Source-record limitation: Source flags the statement as unrecoverable: The official report confirms that this extraction repeats the fixed-parameter tractability question represented canonically by OWR-14298804-007. No separate clean statement is promoted; the original record remains publicly visible and retained for provenance.

Related catalogue records: 30006105. Separate source IDs are preserved for their provenance. The source text below is rendered literally where necessary; damaged formulas and composite extracts are not silently repaired into a new mathematical claim. The supplied catalogue marks this record as unrecoverable. Its extract and terminology are documented, but a complete original problem has not been reconstructed.

Statement recorded in the supplied source.
We ask if the Zonotope Containment Problem is fixed-parameter tractable with respect to the dimension d. In other words: Is there an algorithm that decides Zonotope Containment with running time poly$n + m$ 
\textbackslash{}
cdot f (d) for an arbitrary function f ? The report passage is retained verbatim as the relevant context for this extracted problem. [S2]
Source status.
The supplied catalogue labels this source record open; that metadata was checked against the supplied snapshot on 2026-10-11. The label is preserved for provenance and is not an independent verification of present open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-22.

Short English statement

The supplied fragment concerns duplicate Dimension-Parameterized Zonotope Containment. Its complete mathematical target is not recovered.

Sources

Preparation and review.
The original source and UnsolvedMath contributors retain attribution for the source extract. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, scope notes and formatting. No new human mathematical authorship or independently verified solution is asserted. Permanent source ID 30006106 and code OWR-14298804-008 checked against the complete supplied registry and current repository. Distinct upstream records, including repeated questions and broad or damaged extracts, are retained for complete coverage of the source CS category. A repeated record is not a new independent conjecture. Complete mathematical-equivalence and current-literature review has not been performed. Related canonical identities: 30006105. Preparation and final review follow contribution prompts C and D, with source gaps disclosed. Structural and rendering checks do not establish a new theorem or open status.

Research attempt

Literature, source context and scope

The full catalogue body, including qualifications and historical source status, is preserved above. The original workshop report was inspected in the locally cached publisher PDF text. This attempt distinguishes the source question from any explicitly restricted model introduced below. Repeated records retain their own source IDs and do not become independent conjectures. Elementary results and attributed literature results are not claimed as new original-target progress.

A separate attempted route for the repeated record

This fragment repeats the dimension-parameterized containment question. The external W[1]-hardness result below materially qualifies its dated literature assessment. We test the common proposal to enumerate all sign regions of generator scalar products and describe that as FPT simply because dimension is fixed. For $n\ge d\ge1$, choose normals $g_i=(1,t_i,\ldots,t_i^{d-1})$ with $t_i=i$. Any $d$ are independent: their determinant is the nonzero Vandermonde product. Let $R(n,d)$ be the number of full-dimensional regions cut out by the central hyperplanes $g_i^Tu=0$.
Proposition: \[ R(n,d)=2\sum_{j=0}^{d-1}\binom{n-1}{j}. \] Thus explicit region enumeration on this family is not bounded by $f(d)n^C$ for any fixed exponent $C$ and finite function $f$.
Proof: In dimension one there are two regions. If $n\le d$ independent hyperplanes are linearly equivalent to coordinate hyperplanes and produce $2^n$ regions, agreeing with the formula using zero binomial coefficients. For $n>d\ge2$, add the last hyperplane. The old hyperplanes restrict to a central arrangement on it in dimension $d-1$, with every $d-1$ restricted normals independent because any $d$ original normals are independent. Each region of the restricted arrangement cuts one old region into two, so $R(n,d)=R(n-1,d)+R(n-1,d-1)$. Induction and Pascal's identity yield the formula. For fixed $d$, it grows as a positive constant times $n^{d-1}$. Given $C$, fix $d>C+1$ and let $n$ increase; the ratio $R(n,d)/n^C$ is unbounded, contradicting any such enumeration bound. ∎
Within each region every sign is fixed, so the segment support expression is linear there. This explains the attractiveness of enumeration, but also its dimension-dependent exponent. The argument counts explicit regions; it is not a lower bound against algorithms that implicitly handle many regions at once.

Verification and remaining gaps

The recurrence and formula were checked for $1\le d\le5$ and $1\le n\le10$; for $n=10,d=3$ the count is 92. The repeated source question is not solved by labeling an XP-style enumeration as FPT, and the external conditional hardness is not reproved here.

References and source attribution

The primary Oberwolfach workshop report, https://doi.org/10.4171/owr/2024/50, was inspected on 11 October 2026. Relevant source passages: 2024/50 pp. 2988--2989, repeated FPT question. Catalogue and contributor attribution remain in the original Sources section. Mathematical arguments labeled as this attempt are proved above; cited external results remain attributed to their authors and are not independently reproved or claimed as our discoveries.

https://arxiv.org/html/2509.22849v3. Froese, Grillo, Hertrich and Stargalla, Parameterized Hardness of Zonotope Containment and Neural Network Verification, Theorem 6.1: non-containment is W[1]-hard in dimension. The theorem and support-function reduction were inspected. This is an external conditional complexity obstruction, not our proof; a containment FPT algorithm would also decide its complement in FPT. The preserved dated catalogue assessment does not incorporate this attributed hardness result.

Integration review.
Codex (GPT-6; exact serving revision not exposed) reviewed the written arguments and expanded layout and source-reference wrappers. Repository metadata and the production web renderer were checked using Python 3.13 and JavaScript with MathJax 3.2.2. This model-assisted review is not independent human proof verification; no PDF compilation or formal proof checking is asserted.

Completion Estimate

COMPLETION: 0% No substantiated new progress toward the entire original target is claimed. The worked results and failed route are delimited above; a known result, illustrative model, or auxiliary obstruction is not counted as original-target completion.

Final status and attribution

Attempt status: unresolved. This describes this research attempt, not a new ruling on the source catalogue's status or on external literature claims. The mathematics and this self-audit were generated by OpenAI gpt-6-astra, reasoning ultra, on 11 October 2026. Codex (GPT-6; exact serving revision not exposed) orchestrated the batch and contributed formatting/repository review. No human mathematical authorship, independent human proof checking, or formal verification is claimed. The historical preparation paragraphs in the reproduced catalogue body concern that body only.

Contribution prompt A was applied to the complete definition and supplied source record, followed by the adversarial checks of prompt B. Local reproduction files are batch011/checks.py, computations.json, and the per-ID audit JSON. They were not public links at preparation time. Numerical checks are finite and do not prove unrestricted results. No LaTeX engine was found locally; compilation was not performed by this research agent.

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CS-432-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

gpt 6 astra ultra attempt on CS-432

Fixed-Parameter Tractability of Zonotope Containment

Definitions and mathematical statement

Shared notation.
Standard number systems are denoted by $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$; index ranges, computational models and specialized notation follow the cited problem source and the definitions below.

The centered zonotope generated by $G$ is $Z(G)=\{Gx:x\in[-1,1]^n\}$. Containment means every point of the first zonotope belongs to the second. Dimension-parameterized runtime has a computable factor depending only on $d$.

Fixed-parameter tractability for parameter $k$ means running time $f(k)N^{O(1)}$, where $N$ is the input length and $f$ is computable. The exponent is independent of $k$.

Statement recorded in the supplied source.
Given $G\in\mathbb R^{d\times n}$ and $H\in\mathbb R^{d\times m}$, can one decide whether the zonotope $Z(G)$ is contained in $Z(H)$ in time $f(d)\operatorname{poly}(n+m)$ for some computable function $f$? [S2]
Scope qualification.
The source's centered/noncentered zonotope convention and real-input computational model are not specified in the short statement.
Source status.
The supplied catalogue labels this record open. Its source-status metadata was checked against the supplied snapshot on 2026-10-11. This import does not independently establish current open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-22.

Short English statement

Decide zonotope containment with dimension confined to the parameter-dependent runtime factor.

Sources

Preparation and review.
Original question attributed to the cited source and UnsolvedMath contributors. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, source inspection and formatting. No new human mathematical authorship or independent proof verification is asserted. Canonical ID 30006105 and source code OWR-14298804-007 checked against the complete supplied registry and existing repository statements; known flagged repetitions and obvious identical conjectures excluded. Full mathematical-equivalence/literature review has not been performed. Statement preparation and final review followed contribution prompts C and D; automated checks verify structure and identity, not open status.

Research attempt

Literature, source context and scope

The full catalogue body, including qualifications and historical source status, is preserved above. The original workshop report was inspected in the locally cached publisher PDF text. This attempt distinguishes the source question from any explicitly restricted model introduced below. Repeated records retain their own source IDs and do not become independent conjectures. Elementary results and attributed literature results are not claimed as new original-target progress.

Literature qualification and a support-test route

The source report uses $Z_0(G)=\sum_i[0,g_i]$, while the catalogue defines the centered $Z_c(G)=G[-1,1]^n$. They satisfy \[ Z_0(G)=\tfrac12\sum_i g_i+\tfrac12 Z_c(G). \] The center displacement matters in containment and cannot be discarded. The later primary paper cited below establishes W[1]-hardness of the source non-containment problem, so a general FPT containment algorithm in its rational-input model would imply $\mathrm{FPT}=\mathrm{W[1]}$. This is attributed literature, not a newly proved resolution here; the original metadata remain preserved. Our attempted simplification is to certify containment from the $2d$ coordinate support inequalities. For the centered convention take \[ G=\binom11,\qquad H=\begin{pmatrix}1/2&1/2\\1/2&-1/2\end{pmatrix}. \]
Proposition: The two coordinate projections of $Z_c(G)$ and $Z_c(H)$ agree, but $Z_c(G)\nsubseteq Z_c(H)$.
Proof: Both coordinate ranges are $[-1,1]$. A point of $Z_c(H)$ has coordinates $((s+t)/2,(s-t)/2)$ with $|s|,|t|\le1$, equivalently $|x+y|\le1$ and $|x-y|\le1$. The point $(1,1)\in Z_c(G)$ violates the first inequality. In support notation, maximizing a linear form over independent segment coefficients gives $h_{Z_c(G)}(u)=\sum_i|u^Tg_i|$; hence at $u=(1,1)$ the two supports are two and one. ∎
This also produces a source-convention example: replace each generator by the pair consisting of it and its negative. Then the noncentered sum of these paired segments equals the original centered zonotope, with center zero. Thus the coordinate-test failure is not an artifact of the convention.

Verification and remaining gaps

The four coordinate supports and the diagonal witness were evaluated exactly. Testing all directions, avoiding dimension-dependent enumeration, and matching the precise input model remain unresolved in this attempt. The published conditional obstruction must not be recast as our unconditional impossibility theorem.

References and source attribution

The primary Oberwolfach workshop report, https://doi.org/10.4171/owr/2024/50, was inspected on 11 October 2026. Relevant source passages: 2024/50 pp. 2988--2989, noncentered zonotope definition. Catalogue and contributor attribution remain in the original Sources section. Mathematical arguments labeled as this attempt are proved above; cited external results remain attributed to their authors and are not independently reproved or claimed as our discoveries.

https://arxiv.org/html/2509.22849v3. Froese, Grillo, Hertrich and Stargalla, Parameterized Hardness of Zonotope Containment and Neural Network Verification, Theorem 6.1: non-containment is W[1]-hard in dimension. The theorem and support-function reduction were inspected. This is an external conditional complexity obstruction, not our proof; a containment FPT algorithm would also decide its complement in FPT. The preserved dated catalogue assessment does not incorporate this attributed hardness result.

Integration review.
Codex (GPT-6; exact serving revision not exposed) reviewed the written arguments and expanded layout and source-reference wrappers. Repository metadata and the production web renderer were checked using Python 3.13 and JavaScript with MathJax 3.2.2. This model-assisted review is not independent human proof verification; no PDF compilation or formal proof checking is asserted.

Completion Estimate

COMPLETION: 0% No substantiated new progress toward the entire original target is claimed. The worked results and failed route are delimited above; a known result, illustrative model, or auxiliary obstruction is not counted as original-target completion.

Final status and attribution

Attempt status: unresolved. This describes this research attempt, not a new ruling on the source catalogue's status or on external literature claims. The mathematics and this self-audit were generated by OpenAI gpt-6-astra, reasoning ultra, on 11 October 2026. Codex (GPT-6; exact serving revision not exposed) orchestrated the batch and contributed formatting/repository review. No human mathematical authorship, independent human proof checking, or formal verification is claimed. The historical preparation paragraphs in the reproduced catalogue body concern that body only.

Contribution prompt A was applied to the complete definition and supplied source record, followed by the adversarial checks of prompt B. Local reproduction files are batch011/checks.py, computations.json, and the per-ID audit JSON. They were not public links at preparation time. Numerical checks are finite and do not prove unrestricted results. No LaTeX engine was found locally; compilation was not performed by this research agent.

Go to contribution →