Using Parametric Transformations Toward Polynomial Kernels for Packing Problems Allowing Overlaps
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Bibliographic record
Abstract
We consider the problem of discovering overlapping communities in networks that we model as generalizations of the Set and Graph Packing problems with overlap. As usual for Set Packing problems, we seek a collection S ′ ⊆ S consisting of at least k sets subject to certain disjointness restrictions. In the r -Set Packing with t -Membership, each element of U belongs to at most t sets of S ′ , while in r -Set Packing with t -Overlap, each pair of sets in S ′ overlaps in at most t elements. For both problems, each set of S has at most r elements. Similarly, both of our Graph Packing problems seek a collection K of at least k subgraphs in a graph G , each isomorphic to a graph H ∈ H . In H -Packing with t -Membership, each vertex of G belongs to at most t subgraphs of K , while in H -Packing with t -Overlap, each pair of subgraphs in K overlaps in at most t vertices. For both problems, each member of H has at most r vertices and m edges, where t , r , and m are constants. Here, we show NP-completeness results for all of our packing problems. Furthermore, we give a dichotomy result for the H -Packing with t -Membership problem analogous to the Kirkpatrick and Hell dichotomy [Kirkpatrick and Hell 1978]. Using polynomial parameter transformations, we reduce the r -Set Packing with t -Membership to a problem kernel with O (( r + 1) r k r ) elements and the H -Packing with t -Membership and its edge version to problem kernels with O (( r + 1) r k r ) and O (( m + 1) m k m ) vertices, respectively. On the other hand, by generalizing [Fellows et al. 2008; Moser 2009], we achieve a kernel with O ( r r k r − t − 1 ) elements for the r -Set Packing with t -Overlap and kernels with O ( r r k r − t − 1 ) and O ( m m k m − t − 1 ) vertices for the H -Packing with t -Overlap and its edge version, respectively. In all cases, k is the input parameter, while t , r , and m are constants.
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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 | 0.002 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.002 |
| Open science | 0.001 | 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 it