Linear Programming and the Worst-Case Analysis of Greedy Algorithms on Cubic Graphs
Bibliographic record
Abstract
We introduce a technique using linear programming that may be used to analyse the worst-case performance of a class of greedy heuristics for certain optimisation problems on regular graphs. We demonstrate the use of this technique on heuristics for bounding the size of a minimum maximal matching (MMM), a minimum connected dominating set (MCDS) and a minimum independent dominating set (MIDS) in cubic graphs. We show that for $n$-vertex connected cubic graphs, the size of an MMM is at most $9n/20+O(1)$, which is a new result. We also show that the size of an MCDS is at most $3n/4+O(1)$ and the size of a MIDS is at most $29n/70+O(1)$. These results are not new, but earlier proofs involved rather long ad-hoc arguments. By contrast, our method is to a large extent automatic and can apply to other problems as well. We also consider $n$-vertex connected cubic graphs of girth at least 5 and for such graphs we show that the size of an MMM is at most $3n/7+O(1)$, the size of an MCDS is at most $2n/3+O(1)$ and the size of a MIDS is at most $3n/8+O(1)$.
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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 | 0.007 | 0.030 |
| Meta-epidemiology (narrow) | 0.003 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.002 | 0.005 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.004 | 0.005 |
| Open science | 0.004 | 0.002 |
| Research integrity | 0.002 | 0.005 |
| Insufficient payload (model declined to judge) | 0.006 | 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".