Graph products revisited: Tight approximation hardness of induced matching, poset dimension and more
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Bibliographic record
Abstract
Graph product is a fundamental tool with rich applications in both graph theory and theo-retical computer science. It is usually studied in the form f(G∗H) where G and H are graphs, ∗ is a graph product and f is a graph property. For example, if f is the independence number and ∗ is the disjunctive product, then the product is known to be multiplicative: f(G∗H) = f(G)f(H). In this paper, we study graph products in the following non-standard form: f((G⊕H) ∗ J) where G, H and J are graphs, ⊕ and ∗ are two different graph products and f is a graph property. We show that if f is the induced and semi-induced matching number, then for some products ⊕ and ∗, it is subadditive in the sense that f((G ⊕ H) ∗ J) ≤ f(G ∗ J) + f(H ∗ J). Moreover, when f is the poset dimension number, it is almost subadditive. As applications of this result (we only need J = K2 here), we obtain tight hardness of approximation for various problems in discrete mathematics and computer science: bipartite induced and semi-induced matching (a.k.a. maximum expanding sequences), poset dimension, maximum feasible subsystem with 0/1 coefficients, unit-demand min-buying and single-minded pricing, donation center location, boxicity, cubicity, threshold dimension and independent pack-ing. 1
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| 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 it