On the Parameterized Approximability of Contraction to Classes of\n Chordal Graphs
Bibliographic record
Abstract
A graph operation that {\\em contracts edges} is one of the fundamental\noperations in the theory of graph minors. Parameterized Complexity of editing\nto a family of graphs by contracting $k$ edges has recently gained substantial\nscientific attention, and several new results have been obtained. Some\nimportant families of graphs, namely the subfamilies of chordal graphs, in the\ncontext of edge contractions, have proven to be significantly difficult than\none might expect. In this paper, we study the \\textsc{$\\cal F$-Contraction}\nproblem, where $\\cal F$ is a subfamily of chordal graphs, in the realm of\nparameterized approximation. Formally, given a graph $G$ and an integer $k$,\n\\textsc{ $\\cal F$-Contraction} asks whether there exists $X \\subseteq E(G)$\nsuch that $G/X \\in \\cal F$ and $|X| \\leq k$. Here, $G/X$ is the graph obtained\nfrom $G$ by contracting edges in $X$. We obtain the following results for the\n\\textsc{ $\\cal F$-Contraction} problem. $(1)$ We show that \\textsc{Clique\nContraction} admits a polynomial-size approximate kernelization scheme\n(\\textsf{PSAKS}). $(2)$ We give a $(2+\\epsilon)$-approximate polynomial kernel\nfor \\textsc{Split Contraction} (which also implies a factor\n$(2+\\epsilon)$-\\FPT-approximation algorithm for \\textsc{ Split Contraction}).\nFurthermore, we show that, assuming \\textsf{ Gap-ETH}, there is no\n$\\left(\\frac{5}{4}-\\delta \\right)$-\\FPT-approximation algorithm for\n\\textsc{Split Contraction}. Here, $\\epsilon, \\delta>0$ are fixed constants.\n$(3)$ \\textsc{Chordal Contraction} is known to be \\WTH. We complement this\nresult by observing that the existing \\textsf{W[2]-hardness} reduction can be\nadapted to show that, assuming \\FPT $\\neq$ \\textsf{W[1]}, there is no\n$F(k)$-\\FPT-approximation algorithm for \\textsc{Chordal Contraction}. Here,\n$F(k)$ is an arbitrary function depending on $k$ alone.\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 | 0.004 | 0.028 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.003 |
| Bibliometrics | 0.001 | 0.003 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.006 | 0.017 |
| Open science | 0.006 | 0.005 |
| Research integrity | 0.003 | 0.007 |
| Insufficient payload (model declined to judge) | 0.011 | 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".