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CS-784 · computer science · mathematics · Algorithms and data structures · Graph theory and combinatorics

Duplicate Fare-Zone Approximation Question

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

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.

$y_e=1$ selects an edge as a fare-zone boundary. For a commodity path $P_i$, the constraint limits how many selected edges lie along it; the objective adds its weight once for each selected edge it traverses.

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 tree fare-zone approximation question represented canonically by OWR-14298804-012. No separate clean statement is promoted; the original record remains publicly visible and retained for provenance.

Related catalogue records: 30006109. 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.
What else can we say about approximation results on trees? Open Problem: Fare Zone Assignment Britta Peis (joint work with Britta Peis, Philipp Pabst, Lennart Kauther, and Sam Fiorini) [S2]
Source status.
The supplied catalogue labels this source record partially solved; 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 Fare-Zone Approximation Question. 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 30006110 and code OWR-14298804-013 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: 30006109. 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.

Catalogue entry

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

Contributions 1

CS-784-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

gpt 6 astra ultra attempt on CS-784

Duplicate Fare-Zone Approximation Question

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.

$y_e=1$ selects an edge as a fare-zone boundary. For a commodity path $P_i$, the constraint limits how many selected edges lie along it; the objective adds its weight once for each selected edge it traverses.

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 tree fare-zone approximation question represented canonically by OWR-14298804-012. No separate clean statement is promoted; the original record remains publicly visible and retained for provenance.

Related catalogue records: 30006109. 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.
What else can we say about approximation results on trees? Open Problem: Fare Zone Assignment Britta Peis (joint work with Britta Peis, Philipp Pabst, Lennart Kauther, and Sam Fiorini) [S2]
Source status.
The supplied catalogue labels this source record partially solved; 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 Fare-Zone Approximation Question. 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 30006110 and code OWR-14298804-013 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: 30006109. 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 congestion-dependent LP obstruction for the repeated question

The source fragment repeats the fare-zone agenda. To investigate whether the obvious LP admits a congestion-independent guarantee, use the precise hard-budget system in the neighboring source contribution. For $k\ge2$, take a star with $k+1$ leaf edges. For each unordered pair of leaves include its two-edge path with weight one and budget one. Every edge lies on exactly $k$ paths and has objective coefficient $k$.
Proposition: The integer optimum is $k$, and the standard relaxation $0\le y_e\le1$ has optimum $k(k+1)/2$. Its gap on this family is $(k+1)/2$.
Proof: Every pair of distinct edges is constrained by $y_e+y_f\le1$, so an integer solution selects at most one edge and selecting one attains value $k$. Assigning $y_e=1/2$ to every edge is fractionally feasible. Summing all pair constraints counts each variable exactly $k$ times, giving $k\sum_e y_e\le\binom{k+1}{2}$. Since the objective is $k\sum_e y_e$, the half-vector attains the resulting upper bound $k(k+1)/2$. ∎
Any rounding assertion promising an integer value at least a fixed fraction of this particular LP optimum therefore requires a ratio growing with $k$. This does not exclude a constant-factor approximation measured against the integer optimum using a stronger relaxation or other information. It also does not show an algorithmic lower bound for bounded, fixed $k$.

Verification and remaining gaps

At $k=3$, all sixteen binary choices on the four-leaf star give integer optimum three; the half-vector has objective six. The original broad approximation question is not resolved. Results for a different model allowing over-budget commodities to drop out are not imported.

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 p. 2992, duplicated fare-zone approximation prompt. 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/2512.19493v3. Hoefer, Kauther, Pabst, Peis and Tran, Fare Zone Assignment on Trees, version 3. The model definition was inspected: commodities may drop out when budgets are exceeded, and revenue uses a subadditive pricing function. This differs from the report constraints requiring every budget to hold. Its approximation results are therefore not transferred here without a proved correspondence.

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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