Approximating Dominating Set on Intersection Graphs of Rectangles and\n L-frames
Bibliographic record
Abstract
We consider the Minimum Dominating Set (MDS) problem on the intersection\ngraphs of geometric objects. Even for simple and widely-used geometric objects\nsuch as rectangles, no sub-logarithmic approximation is known for the problem\nand (perhaps surprisingly) the problem is NP-hard even when all the rectangles\nare "anchored" at a diagonal line with slope -1 (Pandit, CCCG 2017). In this\npaper, we first show that for any $\\epsilon>0$, there exists a\n$(2+\\epsilon)$-approximation algorithm for the MDS problem on\n"diagonal-anchored" rectangles, providing the first $O(1)$-approximation for\nthe problem on a non-trivial subclass of rectangles. It is not hard to see that\nthe MDS problem on "diagonal-anchored" rectangles is the same as the MDS\nproblem on "diagonal-anchored" L-frames: the union of a vertical and a\nhorizontal line segment that share an endpoint. As such, we also obtain a\n$(2+\\epsilon)$-approximation for the problem with "diagonal-anchored" L-frames.\nOn the other hand, we show that the problem is APX-hard in case the input\nL-frames intersect the diagonal, or the horizontal segments of the L-frames\nintersect a vertical line. However, as we show, the problem is linear-time\nsolvable in case the L-frames intersect a vertical as well as a horizontal\nline. Finally, we consider the MDS problem in the so-called "edge intersection\nmodel" and obtain a number of results, answering two questions posed by Mehrabi\n(WAOA 2017).\n
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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 | 0.002 | 0.009 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.001 | 0.003 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.003 | 0.005 |
| Open science | 0.003 | 0.003 |
| Research integrity | 0.001 | 0.002 |
| 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".