Researchers presented the SSLD algorithm for optimizing graph coloring

Researchers presented SSLD (Semidefinite Spectral Learning with DSATUR) — a new data preprocessing method that improves the efficiency of the classic heuristic DSATUR algorithm for solving the graph coloring problem (GCP). The method is based on using semidefinite programming (SDP) to select the first color class before running the main algorithm. Unlike the standard approach, SSLD fixes the initial set of vertices, which allows for a significant improvement in the quality of the final graph coloring, minimizing the total number of colors used. The algorithm's effectiveness was tested on more than 1600 benchmarks, including Erdős — Rényi, Watts — Strogatz, and Barabási — Albert graphs, as well as on frequency assignment and production planning tasks. The results show that SSLD either outperforms or matches the standard DSATUR in all tested cases. The main limitation of the method is computational complexity: the preprocessing process makes the algorithm run approximately 195 times slower than the original DSATUR. Nevertheless, the authors confirmed that using SDP-guided selection of the first color class is a promising direction for further optimizing NP-hard problems. This work, published under the arXiv license, demonstrates the potential of hybrid approaches that combine classic heuristics with mathematical programming methods to solve complex combinatorial problems.