Toward a 6/5 Bound for the Minimum Cost 2-Edge Connected Spanning Subgraph
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Bibliographic record
Abstract
Given a complete graph $K_{n}=(V, E)$ with nonnegative edge costs $c\in {\mathbb R}^{E}$, the problem 2EC is that of finding a 2-edge connected spanning multisubgraph of $K_{n}$ of minimum cost. The integrality gap $\alpha\text{2{\it EC}}$ of the linear programming relaxation $\text{2{\it EC}}^{\text{LP}}$ for 2EC has been conjectured to be $\frac{6}{5}$, although currently we only know that $\frac{6}{5}\leq\alpha\text{2{\it EC}}\leq\frac{3}{2}$. In this paper, we explore the idea of using the structure of solutions for $\text{2{\it EC}}^{\text{LP}}$ and the concept of convex combination to obtain improved bounds for $\alpha\text{2{\it EC}}$. We focus our efforts on a family $J$ of half-integer solutions that appear to give the largest integrality gap for $\text{2{\it EC}}^{\text{LP}}$. We successfully show that the conjecture $\alpha\text{2{\it EC}} = \frac{6}{5}$ is true for any cost functions optimized by some $x^{*}\in J$.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.003 | 0.000 |
| Scholarly communication | 0.003 | 0.001 |
| Open science | 0.003 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it