Toward a 6/5 Bound for the Minimum Cost 2-Edge Connected Spanning Subgraph
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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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.005 | 0.023 |
| Meta-epidemiology (narrow) | 0.003 | 0.002 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.003 | 0.003 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.006 | 0.011 |
| Open science | 0.005 | 0.006 |
| Research integrity | 0.004 | 0.012 |
| Insufficient payload (model declined to judge) | 0.018 | 0.005 |
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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".