Fast Low-Rank Maximum-Weight Assignment

Research-attempt provenance

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.

A perfect matching of a complete bipartite graph selects one incident edge at every vertex, maximizing the sum of selected entries of $W$. Rank $r$ is the dimension of the weight matrix's column space.

The rank of a matrix is the dimension of its column space. Matrix completion reconstructs a low-rank matrix from a subset of its entries; matrix sensing reconstructs it from specified linear measurements. Tensor decomposition represents a multiway array as a sum of simple outer products.

Statement recorded in the supplied source.

For the maximum-weight assignment problem on a complete bipartite graph with weight matrix $W\in\mathbb R^{n\times n}$ of rank $r$, can the optimum be computed in time $O(nr^2\log n)$? [S2]

Scope qualification.

For real entries the input-access and arithmetic-cost model must be specified; the question's stated bound is retained.

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 2026-08-21.

Short English statement

Find maximum-weight assignments in the stated rank-dependent near-linear running time.

Sources

• S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30003992 (OWR-16633-009). Catalogue curation and metadata: CC BY 4.0. The supplied source question is retained; definitions and the English summary are editorial additions. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Combinatorial Optimization $2019$, Oberwolfach Reports, DOI 10.4171/owr/2018/50, pp. 2969--3023 https://doi.org/10.4171/owr/2018/50. Underlying Oberwolfach Reports material retains its source-specific licence, including CC BY-SA 4.0 where applicable; see THIRD_PARTY_NOTICES.md.

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 30003992 and source code OWR-16633-009 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.

Literature and context check

The original report, p. 3012, explicitly gives the rank-one sorting observation and asks about rank two and beyond. The representation of the real weight matrix is not fixed. The attempt separates an entry-query obstruction from a concrete failure of extending one-coordinate sorting to factored rank two.

Research attempt: A rank-one access obstruction and a rank-two sorting failure

In a deterministic entry-query model, an unknown rank-one matrix need not reveal its factors quickly. Consider nonnegative matrices with exactly one positive entry and all others zero; every such matrix has rank exactly one. Suppose an algorithm has received zero on $q<n^2-n$ queried locations and now outputs a perfect matching $P$. More than $n$ locations are unqueried, while $P$ contains exactly $n$ locations. Choose an unqueried location outside $P$ and put a one there, zero elsewhere. This matrix is compatible with every query response, has rank one, gives the reported matching weight zero, and has optimum one because the special edge can be extended to a perfect matching. Hence no exact algorithm can always terminate with a correct matching after fewer than $n^2-n$ queries on this adversarial transcript. In particular the requested near-linear rank-one bound cannot hold for this unrestricted entry-query representation.

This is not an obstruction when factors are supplied. If $W=uv^T$, sort both vectors in increasing order and match corresponding entries. Any assignment with $u_i\le u_j$ but assigned $v_a>v_b$ improves or stays equal after swapping, since the gain is $(u_j-u_i)(v_a-v_b)\ge0$. Repeated removal of inversions yields the sorted matching, proving optimality even with negative entries and ties. Factor access permits the report's $O(n\log n)$ comparison bound.

A straightforward rank-two extension, sorting points by their first coordinate, fails. Use row factors $u_1=(0,M)$, $u_2=(1,0)$ and column factors $v_1=(0,1)$, $v_2=(1,M)$, with $M\ge2$. Both first-coordinate orders are $1,2$, so that rule picks the diagonal of \[ W=\begin{pmatrix}M&M^2\\0&1\end{pmatrix}. \] Its value is $M+1$, while the other matching has value $M^2>M+1$ for $M\ge2$. The determinant is $M\ne0$, so this is genuinely rank two. The difference in values can grow quadratically in $M$. This refutes that fixed-coordinate rule, not all geometric rank-two algorithms or all possible projections chosen adaptively from the data.

Verification and adversarial audit

The query adversary uses matrices of exact rank one, avoiding a zero-matrix promise violation. A positive singleton edge always extends to a full matching in the complete bipartite graph. The rank-two determinant and both matching values are explicit; sorting ties are not involved in the first coordinates.

Audit corrections.

The access model is explicit, and the adversarial transcript is used only while compatible with the exact-rank promise. A failed coordinate sort is not marketed as a lower bound for every rank-two algorithm.

External computation.

None; symbolic arguments and boundary checks only.

Scope and remaining gaps

The original target is representation-sensitive. An entry-query version has the proved obstruction, but the factored low-rank version intended by the rank-one observation remains unresolved beyond the demonstrated failed sorting rule. No full negative answer under all input models is claimed.

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% The calculations below the preserved source statement investigate a concrete route, but establish no substantiated new progress toward the entire original target. Known special cases and failed sufficient conditions are not credited as a solution.

References

The original source references and catalogue attribution are preserved in the Sources section above. The following URLs were actually consulted for this attempt; retention of an additional original reference does not claim fresh consultation of it.

• https://ems.press/content/serial-article-files/46772 (consulted 2026-10-11).

Model and contributor disclosure

OpenAI gpt-6-astra, requested reasoning configuration ultra, generated this research attempt on 2026-10-11. The exact serving revision is not exposed. Codex GPT-6 performed orchestration, repository-format review, and adversarial mathematical review of the written argument; its exact serving revision is also not exposed. The mathematical route, boundary checks and final adversarial self-audit followed contribution prompts A then B in docs/contribution_prompts.md, after reading CONTRIBUTING.md. Python 3.14.2 assembled the standalone documents; any mathematical computation is specified in Verification. No theorem prover was used. No human mathematical authorship, substantial human proof correction, or independent human proof checking is asserted. Model-assisted review, source retrieval, and structural checks do not establish correctness or novelty. Full private reasoning transcripts are unavailable as reproducibility artifacts; the complete arguments, inputs and outcomes actually used are stated here or in the companion batch audit.

Final status

Unresolved as a new research contribution. This is a concrete route investigation, not a claim of new complete resolution or a current-status certification. A known answer reproduced for context does not become a new contribution. The catalogue's source-status label is preserved; this attempt status does not assert that a source-labelled solved question is open.

Literature report on TOP-30003992

**A published conditional answer to the general-rank target.** Basu's original problem [1, p. 3012] observes rank-one sorting in $O(n\log n)$, asks about rank two, and proposes $O(nr^2\log n)$ for rank $r$. The current archive statement [3] retains that ambition.

Rohatgi [2, p. 12:3] explicitly discusses Basu's question immediately before Theorem 2 and answers it negatively under the Orthogonal Vectors Conjecture (OVC). In the theorem's notation, there is a constant $c>0$ such that an $O(n^{2-\varepsilon})$ exact minimum-assignment algorithm, for some constant $\varepsilon>0$ and rank function $r(n)=\Omega(c^{\log^* n})$, would refute OVC. Thus the obstruction already occurs at a slowly growing, nonconstant rank; a uniform $O(nr^2\log n)$ bound would be truly subquadratic in that regime. Negating the weight matrix converts maximum to minimum assignment and preserves rank.

**Representation and computational scope.** Appendix A, Theorem 18, reduces bichromatic closest pair to low-rank minimum assignment using transformed, polynomially bounded integer coordinates and **squared** Euclidean costs. For transformed coordinates $a_i,b_j\in\mathbb R^D$, the identity

$$M_{ij}=\lVert a_i\rVert^2+\lVert b_j\rVert^2-2a_i^Tb_j$$

provides a factorization with $D+2$ columns. The geometric construction therefore supplies factors; this is not merely an obstruction caused by reading an arbitrary dense matrix. The appendix's proof displays a polynomial dependence on dimension, which must be retained in runtime transfers. The conclusion uses the paper's fine-grained computational assumptions and bounded-coordinate reductions. It does not establish hardness for arbitrary unit-cost operations on unrestricted reals.

The source defines assignment by its optimum **cost**. An optimizing permutation also supplies that cost by summing its selected entries with the given factors. This should be distinguished from the separate entry-query matching-output argument in CS-354-A1 [4]. That earlier exploration discusses rank-one access and rank-two coordinate sorting, but does not cite Rohatgi or record this OVC obstruction.

**Suggested archive correction.** Add Rohatgi's Theorem 2 and Appendix A as a known conditional negative answer to the uniform general-rank target, with the representation and model qualifications above. Keep the fixed-rank-two exact question separate: this theorem does not resolve it, and this report does not certify its exhaustive current literature status. The mathematical result is Rohatgi's; the contribution here is correction of the archive's literature record, with no claim of new hardness, a new algorithm, or full unconditional disproof.

Original contributor disclosures

Transcribed from the original public submission. Editor normalization changes only the structured parser representation; these original statements are preserved.

Who did the work

LLM only (people only prompted or submitted)

Models used

OpenAI GPT 6.1 Sol, OpenAI GPT 6 Astra

Thinking level / time

Mixed (specified per model below)

Other LLMs

_No response_

Per-model versions, thinking and roles

OpenAI gpt-6-astra | version: unknown (logged model identifier gpt-6-astra) | thinking: medium | time: unknown | role: original literature investigation and research material generation; separate parent context reviewed that material.

OpenAI gpt-6.1-sol | version: unknown (logged model identifier gpt-6.1-sol) | thinking: medium, then high | time: unknown | role: submission preflight at medium; source recheck, concise literature-report assembly and submission preparation at high; a separate fresh context at high independently checked this report and its cited source correspondence (not its own draft).

Model eligibility and verification evidence

The original Medium-setting work was checked by a separate, fresh OpenAI gpt-6.1-sol context at **High**, a listed qualifying model/setting. Provider, model identifier and effort were verified from that review session's own metadata; exact serving revision and time budget are unknown. The reviewer did not draft this report. No previously demonstrated multi-agent architecture is claimed.

The independent check returned **PASS for mathematical/source correspondence**: Basu's complete original question; Rohatgi's literal Theorem 2 and OVC statement; Appendix A/Theorem 18, squared costs, explicit factors and dimension overhead; maximum/minimum and cost/permutation distinctions; fixed-rank-two limitations; and the absence of this citation in the earlier archive contribution. The reviewer independently refetched the primary sources linked in References and matched their bytes to those inspected for this report. This check does not re-prove all external hardness results or certify an exhaustive literature search.

For an exact scope identifier, the reviewed Write-up text (UTF-8, from its first paragraph through its final paragraph, one final newline) has SHA-256 `0dad456a5f3ac69a9d82f8cc58317b35480ca87f6a9753d2a23b5c7986c958b1`. This field records the public review summary; raw session logs and private research notes are not submitted.

People and roles

@yuningyang19: initiated the investigation, authorized this single-topic public literature report, and requested an independent eligible-model review before submission. No human mathematical verification is claimed.

References

[1] Amitabh Basu, “Open Problem: Maximum weighted assignment problem,” in *Combinatorial Optimization*, Oberwolfach Report 50/2018, p. 3012. DOI. Used: the complete original short problem, including rank-one sorting, rank two, and the general-rank target.

[2] Dhruv Rohatgi, *Conditional Hardness of Earth Mover Distance*, APPROX/RANDOM 2019, LIPIcs 145, 12:1–12:17. DOI. Used: the OVC statement in §1.1.1, the discussion and Theorem 2 on p. 12:3, and Appendix A / Theorem 18 and its reduction. This is a source-correspondence report, not independent re-verification of every external theorem used by the paper.

[3] we-solve, TOP-30003992 problem page and official prompt, checked 19 October 2026. Used: canonical handle, statement, input-model qualification, and current contribution listing.

[4] we-solve, CS-354-A1, revision r1, 11 October 2026. Complete text checked. Used: comparison with the recorded entry-query and sorting exploration; its advertised model configuration is not evidence of review of this report.

Declarations

- [X] I have the right to publish this material under CC BY 4.0 (software Apache-2.0), and third-party material is credited.
- [X] I understand that this issue and its contents are public immediately.
- [X] Every LLM and person involved is listed, with versions or "unknown".