New Lower Bounds on the Stability Number of a Graph
Bibliographic record
Abstract
Given a simple, undirected graph G, Motzkin and Straus [Canadian Journal of Mathematics, 17 (1965), 533–540] established that the reciprocal of the stability num-ber of G (the size of the maximum stable set of G) is given by the minimum value of a certain quadratic function over the unit simplex. We propose two new lower bounds on the stability number of G based on this formulation. The first lower bound is ob-tained by minimizing the same objective function over the largest inscribed ball in the unit simplex. Using the fact that quadratic optimization over a full-dimensional ball admits a tight semidefinite programming relaxation, our lower bound can be computed to within any arbitrary precision in polynomial time. For regular graphs, we estab-lish that this lower bound has a closed form solution and that it is tighter than some other existing lower bounds. The second lower bound improves upon the first lower bound and is obtained by a further refinement of the optimal solution that yields the first bound. We evaluate the new bounds and compare them with several other known lower bounds on the DIMACS collection of clique problems. Our computational results reveal that especially the improved lower bound is tighter than all other lower bounds on the majority of the instances. Key words: Maximum stable set, maximum clique, stability number, clique number, semidefinite programming.
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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 | 0.004 | 0.024 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.004 | 0.003 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.003 | 0.008 |
| Open science | 0.004 | 0.003 |
| Research integrity | 0.002 | 0.005 |
| Insufficient payload (model declined to judge) | 0.009 | 0.001 |
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".