HyColor: An Efficient Heuristic Algorithm for Graph Coloring
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Bibliographic record
Abstract
The graph coloring problem (GCP) is a classic combinatorial optimization problem that aims to find the minimum number of colors assigned to the vertices of a graph such that no two adjacent vertices receive the same color. GCP has been extensively studied by researchers from various fields, including mathematics, computer science, and biological science. Due to the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {NP}$</tex-math> </inline-formula>-hard nature, many heuristic algorithms have been proposed to solve GCP. However, existing GCP algorithms focus on either small hard graphs or large-scale sparse graphs (with up to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$10^{7}$</tex-math> </inline-formula> vertices). This article presents an efficient hybrid heuristic algorithm for GCP, named <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">HyColor</small>, which excels in handling large-scale sparse graphs while achieving impressive results on small dense graphs. The efficiency of <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">HyColor</small> comes from the following three aspects: 1) a local decision strategy to improve the lower bound on the chromatic number; 2) a graph-reduction strategy to reduce the working graph; and 3) a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$</tex-math> </inline-formula>-core and mixed degree-based greedy heuristic for efficiently coloring graphs. <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">HyColor</small> is evaluated against three state-of-the-art GCP algorithms across four benchmarks, comprising three large-scale sparse graph benchmarks and one small dense graph benchmark, totaling 209 instances. The results demonstrate that <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">HyColor</small> consistently outperforms existing heuristic algorithms in both solution accuracy and computational efficiency for the majority of instances. Notably, <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">HyColor</small> achieved the best solutions in 194 instances (over 93%), with 34 of these solutions significantly surpassing those of other algorithms. Furthermore, <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">HyColor</small> successfully determined the chromatic number and achieved optimal coloring in 128 instances.
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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 | 0.005 | 0.000 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.002 | 0.001 |
| Scholarly communication | 0.003 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.001 | 0.001 |
| 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