Approximating Minimum Sum Coloring with Bundles
Bibliographic record
Abstract
In the Minimum Sum Coloring with Bundles problem, we are given an undirected graph G = (V,E) and (not necessarily disjoint) bundles V_1, V_2, …, V_p ⊆ V with associated weights w_1, …, w_p ≥ 0. The goal is to give a proper coloring of G using positive integers to minimize the weighted average/total completion time of all bundles, where the completion time of a bundle is the maximum integer assigned to one of its nodes. This is a common generalization of the classic Minimum Sum Coloring problem, i.e. when all bundles are singleton nodes, and the classic Chromatic Number problem, i.e. the only bundle is all of V. Despite its generality as an extension of Minimum Sum Coloring, only very special cases have been studied with the most common being the line graph L(H) of a graph H (also known as Coflow Scheduling). We provide the first constant-factor approximation in perfect graphs and, more generally, graphs whose chromatic number is within a constant factor of the maximum clique size in any induced subgraph. For example, we obtain constant-factor approximations for graphs that are well-studied in minimum sum coloring such as interval graphs and unit disk graphs. Next, we extend our results to get constant-factor approximations for a general model where the bundles are disjoint (i.e. can be thought of as jobs brought by the corresponding client) and we are only permitted to color/schedule a bounded number of jobs from each bundle at any given time. Specifically, we get constant-factor approximations for this model if the nodes of graph G have an ordering v_1, v_2, …, v_n such that the left-neighborhood N_𝓁(v_i) : = {v_j : j < i, v_iv_j ∈ E} can be covered by O(1) cliques. For example, this applies to chordal graphs, unit disc graphs, and circular arc graphs.
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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 | 0.001 | 0.007 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.003 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| 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".