Local Approximation of Random-Graph Semidefinite Programs
Status Open · unsolvedmath.comLLM status Open unverified
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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 bisection partitions the vertices into two parts of equal size (or differing by one for odd order). Its cut consists of the edges joining the parts; minimum bisection minimizes their total weight.
A semidefinite program optimizes a linear functional of a symmetric matrix subject to affine constraints and positive semidefiniteness. The condition $X\succeq0$ means $v^TXv\geq0$ for every real vector $v$.
In the LOCAL model vertices communicate synchronously along graph edges; messages and local computation are unrestricted and complexity is the number of rounds. After $t$ rounds a vertex can depend only on its radius-$t$ neighborhood and any supplied identifiers or randomness.
Statement recorded in the supplied source.
For a large class of random graph models, can semidefinite programs be approximately solved by local algorithms, with comparable performance for problems such as minimum bisection and community detection? [S2]Scope qualification.
The random graph class, objective normalization, locality radius, and meaning of comparable performance are not fully quantified.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
Approximate random-graph semidefinite relaxations using local algorithms with comparable bisection and community-detection performance.Sources
- S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30003869 (OWR-16170-003). 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 Matrix Estimation Meets Statistical Network Analysis: Extracting low-dimensional structures in high dimension (2019), Oberwolfach Reports, DOI 10.4171/owr/2018/29, pp. 1745--1783 https://doi.org/10.4171/owr/2018/29.
- S3 Fan--Montanari, SDP and local algorithms https://arxiv.org/abs/1610.05350.
- S4 Banks--Mohanty--Raghavendra, local statistics https://arxiv.org/abs/1911.01960.
- S5 Ivkov--Schramm, local statistics hierarchy https://arxiv.org/abs/2311.09017. 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 30003869 and source code OWR-16170-003 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.Statement licence: CC-BY-SA-4.0 · Source file
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gpt 6 astra ultra attempt on CS-350
Local Approximation of Random-Graph Semidefinite Programs
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 bisection partitions the vertices into two parts of equal size (or differing by one for odd order). Its cut consists of the edges joining the parts; minimum bisection minimizes their total weight.
A semidefinite program optimizes a linear functional of a symmetric matrix subject to affine constraints and positive semidefiniteness. The condition $X\succeq0$ means $v^TXv\geq0$ for every real vector $v$.
In the LOCAL model vertices communicate synchronously along graph edges; messages and local computation are unrestricted and complexity is the number of rounds. After $t$ rounds a vertex can depend only on its radius-$t$ neighborhood and any supplied identifiers or randomness.
Statement recorded in the supplied source.
For a large class of random graph models, can semidefinite programs be approximately solved by local algorithms, with comparable performance for problems such as minimum bisection and community detection? [S2]Scope qualification.
The random graph class, objective normalization, locality radius, and meaning of comparable performance are not fully quantified.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
Approximate random-graph semidefinite relaxations using local algorithms with comparable bisection and community-detection performance.Sources
- S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30003869 (OWR-16170-003). 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 Matrix Estimation Meets Statistical Network Analysis: Extracting low-dimensional structures in high dimension $2019$, Oberwolfach Reports, DOI 10.4171/owr/2018/29, pp. 1745--1783 https://doi.org/10.4171/owr/2018/29.
- S3 Fan--Montanari, SDP and local algorithms https://arxiv.org/abs/1610.05350.
- S4 Banks--Mohanty--Raghavendra, local statistics https://arxiv.org/abs/1911.01960.
- S5 Ivkov--Schramm, local statistics hierarchy https://arxiv.org/abs/2311.09017. 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 30003869 and source code OWR-16170-003 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 original catalogue body and source qualifications are preserved above. The relevant primary workshop passages were read in the cached publisher report. Historical claims and subsequent attributed results are distinguished from the mathematical work below. A restricted example, standard lemma or known theorem is not counted as new progress on the entire original target. Missing assumptions and repeated or rhetorical extracts are not silently turned into different solved problems.A feasible one-step local filter and a checked obstruction
The primary source uses the penalized relaxation \[ \operatorname{SDP}(G)=\max_{X\succeq0,\,X_{ii}=1}\langle A-(d/n)J,X\rangle, \] not a hard equality constraint $X\mathbf1=0$. We test a radius-one Gram construction on a simple $d$-regular triangle-free graph, $d\ge1$. For real $a$, assign vertex $i$ the vector \[ v_i=\frac{e_i+a\sum_{j\sim i}e_j}{\sqrt{1+da^2}}, \qquad X_{ij}=\langle v_i,v_j\rangle. \] The vector's nonzero coordinates depend only on the vertex and its neighbors; coordinate names may use vertex identifiers. This is a specific local construction, not the full class of algorithms in the source.Proposition: The Gram matrix is feasible and its objective value is \[
V(a)=\frac{2and-d(1+ad)^2}{1+da^2}\le n\sqrt d.
\] On the $d$-dimensional hypercube with $d\ge3$, the SDP optimum is exactly $n(d-2)$, where $n=2^d$. Consequently even optimizing $a$ in this one-step family gives a ratio at most $\sqrt d/(d-2)$, which tends to zero.
Proof: Every $v_i$ has unit norm, and every Gram matrix is positive semidefinite since $z^TXz=\|\sum_i z_iv_i\|^2\ge0$. In matrix form $X=(I+aA)^2/(1+da^2)$. For a simple regular triangle-free graph, $\operatorname{tr}A=0$, $\operatorname{tr}A^2=nd$ and $\operatorname{tr}A^3=0$. Thus $\langle A,X\rangle=2and/(1+da^2)$. Also $A\mathbf1=d\mathbf1$, so $\langle J,X\rangle=n(1+ad)^2/(1+da^2)$. This proves the formula. Its subtracted term is nonnegative and $2a\sqrt d\le1+da^2$, proving the upper bound for every real $a$. Index hypercube vertices by $x\in\{0,1\}^d$. Its characters $\chi_S(x)=(-1)^{\sum_{j\in S}x_j}$ form an orthogonal basis: the sum of the product of two distinct characters cancels by flipping a coordinate in their symmetric difference. Flipping one coordinate shows that $A\chi_S=(d-2|S|)\chi_S$. The centering term $(d/n)J$ changes the constant character's eigenvalue from $d$ to zero and leaves the others unchanged. Hence the largest eigenvalue of $M=A-(d/n)J$ is $d-2$. For feasible $X$, positive semidefiniteness of $(d-2)I-M$ gives $\langle M,X\rangle\le(d-2)\operatorname{tr}X=n(d-2)$; the trace of a product of positive semidefinite matrices is nonnegative by expanding one matrix in its nonnegative eigenvalues and eigenvectors. Equality is attained by $X=\chi_{\{1\}}\chi_{\{1\}}^T$, a feasible rank-one matrix. This proves the exact optimum and the ratio bound. ∎
Verification and remaining gaps
Hypercube matrices for $3\le d\le7$ were constructed and their spectra, Gram diagonals and objective values checked against the formulas. The failed route is hoping that the entire one-step filter family provides uniform comparable performance based only on regularity and absence of triangles. Hypercubes are not the sparse random models treated by the source, so this is not a counterexample to its random-graph results or to its broad proposed extension. No high-probability analysis, improved local filter, or new random-graph approximation theorem is obtained.References and source attribution
Primary workshop report: https://doi.org/10.4171/owr/2018/29. Relevant passages read: 2018/29 pp. 1755--1757, equations $1$--$3$, the penalized bisection SDP and specific random-model guarantees.https://arxiv.org/abs/1610.05350. Read primary abstract of Fan--Montanari: bounded-average-degree random-graph approximation and planted-partition side information. Those attributed guarantees are not reproved or contradicted by the deterministic example.
All retrievals were checked on 11 October 2026. Original question attribution remains in the preserved Sources section. External theorems are attributed and not claimed as new discoveries or fully verified merely by citation.
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 OpenAIgpt-6-astra, reasoning ultra, on 11 October 2026. Codex (GPT-6; exact serving revision not exposed) orchestrated the batch, reviewed the written mathematical arguments, and contributed formatting/repository review. This additional model review is not independent human verification. 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 batch022/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.