Better-Than-Two Approximation for Multiplicative Connectivity Augmentation

Repeated augmentation question: an exact star case and a failure of minimum-cut-only coverage

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.

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 $\rho<2$.

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 is a duplicate extraction of the multiplicative connectivity augmentation problem retained in OWR-8415356-018. No separate clean mathematical statement is supplied; this source ID preserves its provenance.

Related catalogue records: 30004907. 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.

Therefore, we propose first studying

\textbackslash{}

alpha-multiplicative connectivity augmentation for

\textbackslash{}

alpha

\textbackslash{}

leq 6/5 which may alternately be called polygon augmentation, following the naming convention of the problems tree augmentation and cactus augmentation in which the goal is to increase the connectivity of a graph by 1. The main open problem is to design a better-than-2 approximation for this problem for any constant

\textbackslash{}

alpha > 1. Open Problem:

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alpha-Multiplicative Connectivity Augmentation and Polygon Augmentation Nathan Klein [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 better-Than-Two Approximation for Multiplicative Connectivity Augmentation. Its complete mathematical target is not recovered.

Sources

• S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30004908 (OWR-8415356-019). Catalogue curation and metadata: CC BY 4.0. The source extract is retained; definitions, English summaries and qualifications 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 Weighted connectivity augmentation https://arxiv.org/abs/2209.07860.
• Source-specific Oberwolfach Reports licences, including CC BY-SA 4.0 where applicable, are retained; see THIRD_PARTY_NOTICES.md.

Verbatim supplied catalogue question

Therefore, we propose first studying

\textbackslash{}

alpha-multiplicative connectivity augmentation for

\textbackslash{}

alpha

\textbackslash{}

leq 6/5 which may alternately be called polygon augmentation, following the naming convention of the problems tree augmentation and cactus augmentation in which the goal is to increase the connectivity of a graph by 1. The main open problem is to design a better-than-2 approximation for this problem for any constant

\textbackslash{}

alpha > 1. Open Problem:

\textbackslash{}

alpha-Multiplicative Connectivity Augmentation and Polygon Augmentation Nathan Klein

Duplicate scope

This fragment repeats the augmentation problem retained as 30004907. The original paragraph on printed page 2945 supplies the base connectivity, weighted candidate links and rounded target. We examine a simple base class where the deficient cuts can be characterized exactly, then test extending the rule of covering only the original minimum cuts. This is a separate attempted route on the same underlying question, not a new independent conjecture.

An exact unit-cost star augmentation.

Let $G$ be a star with $h\ge2$ leaves. Its edge connectivity is one. Suppose every pair of leaves is available as a unit-cost link, and $1<\alpha\le2$, so the target is $\lceil\alpha\rceil=2$. A selected set of links is feasible exactly when each leaf is incident to at least one link. To prove this, represent any cut by its side not containing the center; that side is a nonempty leaf subset $S$. Its original cut size is $|S|$. If $|S|\ge2$, the target is already met. If $S$ is a singleton, it needs exactly one crossing link, equivalent to that leaf's having positive link degree. This proves necessity and sufficiency for every cut.

The optimum number of links is $\lceil h/2\rceil$. Every link covers at most two leaves, giving this lower bound. For even $h$, pair all leaves and take the resulting $h/2$ links. For odd $h\ge3$, pair $h-1$ leaves and connect the remaining leaf to any paired leaf; this uses $(h+1)/2$ links and covers every leaf. The construction is polynomial and exact, but uses the complete candidate-link set, unit costs and the target two. It does not solve arbitrary weighted augmentation.

Original minimum cuts need not be the only deficient cuts.

Keep a star with four leaves, but now take $\alpha=3$, so the target is three. Select two parallel links between leaves one and two, and two between leaves three and four. Each singleton leaf cut increases from one to three, so all original minimum cuts meet the target. The cut with side consisting of leaves one and two still has only its two star edges: none of the selected links crosses it. Its capacity is two, below target. Thus covering every original minimum cut to its required new size is not sufficient for general multiplicative augmentation. All originally sub-target cuts matter, including those that were not minimum cuts. The source allows candidate links as a multiset, so the parallel links in this countercheck respect that model.

This example uses $\alpha=3$, outside the source's proposed initial polygon range $\alpha\le6/5$. It tests the full "any constant greater than one" goal, not an impossibility claim for that smaller range. Conversely the exact target-two star construction lies within the smaller range when $\alpha\le6/5$ because the ceiling is still two. Neither observation transfers its algorithm to arbitrary large base connectivity.

Audit

The primary augmentation paragraph and the duplicate link were inspected. The audit checked the ceiling at base connectivity one, both sides of every star cut, the odd-leaf construction and the scope of the parallel-link example. All proofs are by explicit cut counts, with no optimization solver used. The missing step is an algorithm coordinating all deficient cuts with arbitrary costs and base graph structure while achieving a uniform factor below two. This remains unresolved under the repeated source record.

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. A complete-link unit-cost star case is solved exactly, but minimum-cut-only coverage fails in the general multiplicative target and no full approximation 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. A complete-link unit-cost star case is solved exactly, but minimum-cut-only coverage fails in the general multiplicative target and no full approximation algorithm is supplied.

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.