Matrix Partitions of Graphs: Algorithms and Complexity
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Bibliographic record
Abstract
Recently, there has been much interest in studying certain graph partitions that generalize graph colourings and homomorphisms. They are described by a pattern, usually viewed asa symmetric ${0, 1, *}$-matrix $M$. Existing results focus on recognition algorithms and characterization theorems for graphsthat admit such $M$-partitions, or $M$-partitions in which vertices of the input graph $G$have lists of admissible parts. For (homomorphism) problems with costs, researchers havealso investigated the approximability of the problem.In this thesis, we study the complexity of these matrix partition problems.First, we investigate the complexity of counting $M$-partitions. The complexity of counting problemsfor graph colourings and graph homomorphisms has been previously classified, and most turned out to be $sharpP$-complete, with only trivial exceptions.By contrast, we exhibit many $M$-partition problems with interesting non-trivial counting algorithms; moreover these algorithms appear to depend on highly combinatorial tools. In fact, our tools are sufficient to classify the complexity of counting$M$-partitions for all matrices $M$ of size less than four.Then, we turn our attention to the homomorphism problems with costs.Previous results include partial classification of approximation complexityfor doubly convex bipartite graphs.We complete these results and extend them to all digraphs.We prove that if $H$ is a co-circular arc bigraph,then the minimum cost graph homomorphism problem to $H$ admits a polynomial time constant ratio approximation algorithm.This solves a problem posed in an earlier paper. Our algorithm is obtained by derandomizinga two-phase randomized procedure. In the final third of the thesis, we present a partial dichotomy forthe complexity of exact minimization of homomorphism costs,when the cost function is a constant across the vertices of the input graph. We show that the dichotomy is complete when the target graph is a tree.
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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 | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| 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