gpt 6 astra ultra attempt on CS-383

Multiplicative augmentation: exact cut deficits and the limits of threshold rounding

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.

In $\alpha$-multiplicative connectivity augmentation the input is a $k$-edge-connected graph and a set of candidate links with nonnegative costs. One selects minimum-cost links so that the augmented graph is at least $\lceil\alpha k\rceil$-edge-connected. Edge connectivity is the minimum number of edges in a nontrivial cut. An approximation factor below two means a polynomial-time algorithm with cost at most $\rho$ times optimum for some $\rho<2$.

Statement recorded in the supplied source.

For every fixed $\alpha>1$, does $\alpha$-multiplicative connectivity augmentation admit an approximation factor strictly below $2$? [S2]

Scope qualification.

The improvement may depend on fixed $\alpha$ but must be uniform over the input instances. Feasibility assumes enough available links to achieve the target.

Source status.

The supplied catalogue labels this record partially solved. 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

Improve the factor-two approximation barrier for increasing edge connectivity by a fixed multiplicative factor.

Sources

• S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30004907 (OWR-8415356-018). 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 $2022$, Oberwolfach Reports, DOI 10.4171/owr/2021/53, pp. 2893--2954 https://doi.org/10.4171/owr/2021/53.
• S3 Combinatorial Optimization, Oberwolfach Reports 53/2021, primary report PDF https://ems.press/content/serial-article-files/46931. Underlying Oberwolfach Reports material retains its source-specific licence, including CC BY-SA 4.0 where applicable; see THIRD_PARTY_NOTICES.md.

Verbatim supplied catalogue question

For every fixed $\alpha>1$, does $\alpha$-multiplicative connectivity augmentation admit an approximation factor strictly below $2$?

The cut-cover route

Put $t=\lceil\alpha k\rceil$. For a nontrivial vertex set $S$, let $d_G(S)=|\delta_G(S)|$ and $b(S)=\max\{0,t-d_G(S)\}$. A selected multiset $F$ of available links is feasible exactly when \[ |F\cap\delta_L(S)|\ge b(S)\quad\hbox{for every nontrivial }S. \] Indeed each selected crossing link adds one to that cut, so its new size is $d_G(S)+|F\cap\delta_L(S)|$. Requiring this to be at least $t$ is exactly the displayed inequality. This uses the source's cut definition of edge connectivity, without an additional disjoint-path theorem. Only cuts with $d_G(S)<t$ matter. Since this cut size is integral, $d_G(S)<t$ implies $d_G(S)<\alpha k$, including when $\alpha k$ is an integer. This explains the relevance of a representation of near-minimum cuts. It does not solve their simultaneous weighted covering problem.

A feasible optimum fractional solution missed by thresholding.

Take $G=K_{k+1}$ for $k\ge1$. A cut with $s$ vertices on one side has $d_G(S)=s(k+1-s)\ge k$, so its edge connectivity is $k$. Let $r=t-k>0$. Provide $q$ parallel candidate copies of every original edge, each of unit cost, where $q$ is chosen so large that $r/(qk)<1/2$. Give every candidate fractional value $x_e=r/(qk)$. For any cut the added fractional capacity is $(r/k)d_G(S)$, and hence the total is $(t/k)d_G(S)\ge t$. Thus all cut constraints hold and all variables lie strictly below one half.

This fractional solution is optimal for the cut relaxation. Each singleton cut needs $r$ added capacity. Summing the $k+1$ singleton constraints counts each link twice, so every feasible fractional solution has cost at least $(k+1)r/2$. The displayed solution has exactly that cost, since there are $qk(k+1)/2$ links. Rounding by retaining only values at least one half selects no link and fails every deficient singleton cut. This example does not claim that the fractional point is an extreme point. An iterative-rounding theorem about extreme optima cannot be refuted by an arbitrary optimum, and this distinction is essential.

Why repeating a one-unit algorithm needs amortization.

Suppose one has a factor-$\rho$ routine for increasing the current connectivity by one. A naive analysis applies it $r=t-k$ times and compares each round's optimum with the cost $C^*$ of one final optimal augmentation. That comparison is valid: at any round, adding the final optimum's as-yet unselected links would attain at least $t$, so in particular attains the next target. Its additional cost is at most $C^*$. But summing these bounds yields only $r\rho C^*$, not $\rho C^*$. For fixed $\alpha>1$, $r$ grows with $k$, so this argument alone does not even give a uniform constant factor. Some stronger charging argument or coordinated multilevel algorithm would be required. This is a failure of the proposed analysis, not a constructed lower bound on every repeated-augmentation routine.

Audit

The primary augmentation problem on printed page 2945 was inspected, including the weighted multiset of links. The audit retained the ceiling, all deficient cuts, and the fact that compactly representing constraints does not automatically yield stronger rounding. The fractional example uses a simple base graph and only the explicitly permitted parallel candidate links. No LP solver or augmentation experiment was run. A factor uniformly below two for the full original class remains unproved here.

Process and contributions

Mathematics and model-assisted adversarial self-audit: OpenAI gpt-6-astra, ultra reasoning, as dispatched for this research batch; exact serving revision not exposed. Codex (GPT-6; exact serving revision not exposed) materially assisted with orchestration and format review. No human mathematical authorship, mathematical edits, or independent human proof checking is asserted. Contribution prompt A was applied to the full definition and supplied source row, then prompt B was applied adversarially to the derived results. The item-specific audit above records the scope checks and rejected extrapolations. Local assembly and structural checks do not establish mathematical correctness. The source catalogue status is preserved in TOP_PROBLEM and is separate from this attempt status.

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 proved material is a rederivation of known or auxiliary facts rather than a substantiated new advance on the original target. Exact cut deficits and two failed rounding/composition arguments are established, but no uniform better-than-two augmentation algorithm is supplied.

References

The complete original Sources subsection above is retained, including catalogue attribution and licence notices. The additional list records primary sources actually consulted for this attempt on 11 October 2026; uncited source links reproduced above were not all rechecked.

• Oberwolfach Report 53/2021, Combinatorial Optimization, original question and surrounding discussion read from the primary PDF text on 11 October 2026. Historical statements are not a current-status survey. https://ems.press/content/serial-article-files/46931.

Final status

Unresolved in this attempt. Exact cut deficits and two failed rounding/composition arguments are established, but no uniform better-than-two augmentation algorithm is supplied.