Using Parametric Transformations Toward Polynomial Kernels for Packing Problems Allowing Overlaps
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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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 | 0.003 | 0.021 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.003 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.003 | 0.013 |
| Open science | 0.003 | 0.006 |
| Research integrity | 0.002 | 0.007 |
| Insufficient payload (model declined to judge) | 0.005 | 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".