Runtime Thresholds for Ant-Colony Optimization
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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.OneMax maps a bit string $x\in\{0,1\}^n$ to $\sum_i x_i$. In ant-colony optimization, sampling probabilities are represented by pheromone values; evaporation updates these values with parameter $\rho$. Expected runtime counts evaluations until an optimum is first reached.
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 exact algorithm, initialization, pheromone bounds, and permitted scaling of $\rho$ with $n$ are source-specific.Source-record limitation: The ant-colony variant, update rule, and initialization are not defined; evaporation parameter alone does not specify the runtime question.
Statement recorded in the supplied source.
Determine expected runtime bounds for ant-colony optimization algorithms on simple optimization problems; in particular, for OneMax classify evaporation factors $\rho$ for which the runtime is polynomial versus exponential. [S2]Source status.
The supplied catalogue labels this source record open; 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 2026-08-21.Short English statement
Find runtime thresholds for ant-colony optimization, especially the evaporation rates separating polynomial and exponential behavior on OneMax.Sources
- S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30000612 (OWR-1386-020). 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 Combinatorics, Probability and Computing (2007), Oberwolfach Reports, DOI 10.4171/owr/2006/48, pp. 2875--2952 https://doi.org/10.4171/owr/2006/48.
- Source-specific Oberwolfach Reports licences, including CC BY-SA 4.0 where applicable, are retained; see THIRD_PARTY_NOTICES.md.
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 30000612 and code OWR-1386-020 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. 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.Statement licence: CC-BY-SA-4.0 · Source file
Contributions 1
gpt 6 astra ultra attempt on CS-648
Runtime Thresholds for Ant-Colony Optimization
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.OneMax maps a bit string $x\in\{0,1\}^n$ to $\sum_i x_i$. In ant-colony optimization, sampling probabilities are represented by pheromone values; evaporation updates these values with parameter $\rho$. Expected runtime counts evaluations until an optimum is first reached.
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 exact algorithm, initialization, pheromone bounds, and permitted scaling of $\rho$ with $n$ are source-specific.Source-record limitation: The ant-colony variant, update rule, and initialization are not defined; evaporation parameter alone does not specify the runtime question.
Statement recorded in the supplied source.
Determine expected runtime bounds for ant-colony optimization algorithms on simple optimization problems; in particular, for OneMax classify evaporation factors $\rho$ for which the runtime is polynomial versus exponential. [S2]Source status.
The supplied catalogue labels this source record open; 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 2026-08-21.Short English statement
Find runtime thresholds for ant-colony optimization, especially the evaporation rates separating polynomial and exponential behavior on OneMax.Sources
- S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30000612 (OWR-1386-020). 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 Combinatorics, Probability and Computing $2007$, Oberwolfach Reports, DOI 10.4171/owr/2006/48, pp. 2875--2952 https://doi.org/10.4171/owr/2006/48.
- Source-specific Oberwolfach Reports licences, including CC BY-SA 4.0 where applicable, are retained; see THIRD_PARTY_NOTICES.md.
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 30000612 and code OWR-1386-020 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. 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 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.Two exact model endpoints, without inventing an evaporation rule
The primary report already states particular ant-colony runtime results, including a high-evaporation relation to the $(1+1)$ evolutionary algorithm and a change near an inverse-linear evaporation scale. The extract does not define the complete pheromone update rule. We therefore cannot interpolate between evaporation factors merely from its symbol $\rho$. We test two explicitly defined sampling processes on OneMax, as auxiliary endpoints rather than an asserted complete ACO model.Proposition: If every evaluation is an independent uniform bit string in $\{0,1\}^n$, the expected number of evaluations through the first optimum is $2^n$. For $n\ge2$, the elitist $(1+1)$ process that flips each bit independently with probability $1/n$ and accepts a child whenever its OneMax value is no smaller than its parent's has expected optimization time at most $enH_n$ mutation evaluations from any initial string, where $H_n=\sum_{j=1}^n1/j$.
Proof: Uniform sampling hits the unique all-ones string with probability $2^{-n}$ independently at every evaluation. Its first-success time has tail $(1-2^{-n})^t$ after $t$ evaluations, so summing the geometric tail gives mean $2^n$. For the elitist process, if the current string has $j>0$ zeros, flipping exactly one of those zeros and no other bit improves the objective. Its probability is \[
p_j=\frac jn(1-1/n)^{n-1}\ge\frac j{en}.
\] To justify the inequality, $-\log(1-1/n)=\int_0^{1/n}(1-t)^{-1}dt\le1/(n-1)$; exponentiating gives $(1-1/n)^{n-1}\ge e^{-1}$. While at level $j$, even if accepted equal-fitness moves change the string, each independent mutation has probability at least $p_j$ of leaving toward a better level. The probability of spending at least $t$ further unsuccessful iterations at that level is therefore at most $(1-p_j)^t$, by conditional multiplication. Its expected residence time is at most $1/p_j\le en/j$. Elitism prevents return to a worse level, and skipping levels only decreases this sum, giving the bound $enH_n$. ∎
Verification and remaining gaps
For $n=2,\ldots,8$, the exact transition probabilities between OneMax levels were computed by summing all bit-flip patterns, and the triangular hitting-time equations were solved with rational arithmetic. Their worst initial-state expectations satisfy the displayed bound. The no-learning uniform sampler has exponential mean, whereas the specified elitist process has a polynomial bound. These processes do not define every intermediate ACO update or identify an evaporation threshold. In particular, assigning the first process the label $\rho=0$ would additionally require a fixed uniform initialization and a rule leaving it unchanged, assumptions not supplied by the original extract. The attempted continuum classification stops at the missing update dynamics, not at a proved threshold.References and source attribution
Primary workshop report: https://doi.org/10.4171/owr/2006/48. Relevant passages read: 2006/48 pp. 2943--2945, the specified simple ant-colony model and runtime regimes already reported.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 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 batch016/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.