Multitasking Capacity: Hardness Results and Improved Constructions
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Bibliographic record
Abstract
We consider the problem of determining the maximal $\alpha \in (0,1]$ such that every matching $M$ of size $k$ (or at most $k$) in a bipartite graph $G$ contains an induced matching of size at least $\alpha |M|$. This measure was recently introduced in [N. Alon et al., Adv. Neural Inf. Process. Syst., 2017, pp. 2097--2106] and is motivated by computational models in cognitive neuroscience as well as by modeling interference in radio and communication networks. We prove various hardness results for computing $\alpha$ either exactly or approximately. En route to our results, we also consider the maximum connected matching problem: determining the largest matching $N$ in a graph $G$ such that every two edges in $N$ are connected by an edge. We prove a nearly optimal $n^{1-\epsilon}$ hardness of approximation result (under randomized reductions) for connected matching in bipartite graphs (with both sides of cardinality $n$). Toward this end we define bipartite half-covers: a new combinatorial object that may be of independent interest. To our knowledge, the best previous hardness result for the maximum connected matching problem was that it is hard to approximate within some constant $\beta>1$. Finally, we demonstrate the existence of bipartite graphs with $n$ vertices on each side of average degree $d$, achieving $\alpha=1/2-\epsilon$ for matchings of size sufficiently smaller than $n/d$. This nearly matches the trivial upper bound of $1/2$ on $\alpha$ which holds for any graph containing a path of length 3.
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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 | 0.000 |
| 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