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 distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.003 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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 teacher head, 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".