Duality for Geometric Set Cover and Geometric Hitting Set Problems on Pseudodisks
Bibliographic record
Abstract
Given an instance of a geometric set cover problem on a set of points X and a set of objects R, the dual is a geo-metric hitting set problem on a set of points P and a set of objects Q, where there exists a one-to-one mapping from each xj ∈ X to a dual object Qj ∈ Q and for each Ri ∈ R to a dual point in pi ∈ P, so that a dual point pi is contained in a dual object Qj if and only if the cor-responding primal point xj is covered by the object Ri. In this work, we explore the setting of geometric dual-ity for geometric set cover problems on pseudodisks. We first show that there does not always exist a geometric dual on pseudodisks. We initiate the search for a char-acterization of the class of objects that may be dualized by identifying a sufficient (but not necessary) property for a dual to exist on distinct pseudodisks, called the pair-cover and crossing-quad free property. We show that such problems may be dualized into hitting set in-stances on pseudodisks by building a planar support for the dual instance, and then constructing an orthogo-nal drawing of the support which we transform into a dual set of pseudodisks. A corollary of these results is a PTAS for dualizable set cover problems using the PTAS for hitting set on pseudodisks. 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 | 0.002 | 0.011 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.004 | 0.008 |
| Open science | 0.002 | 0.004 |
| Research integrity | 0.003 | 0.005 |
| Insufficient payload (model declined to judge) | 0.008 | 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".