Preprocessing method SSLD improves graph coloring heuristic DSATUR across 1,600+ instances

Researchers from the University of Lorraine propose SSLD (Semidefinite Spectral Learning with DSATUR), a method that improves the DSATUR graph coloring heuristic by preprocessing a first color class using semidefinite programming (SDP). The Graph Coloring Problem (GCP) is NP-hard, and DSATUR is one of the fastest heuristics despite typically using more colors than state-of-the-art algorithms.

SSLD’s preprocessing step leverages an SDP similar to one used for computing the Lovász theta number, a well-known graph invariant. The authors state that SSLD is the first approach to improve DSATUR by preprocessing through fixed color classes.

The evaluation covers over 1,600 benchmark instances, including DIMACS datasets, random graphs (Erdős–Rényi, Watts–Strogatz, Barabási–Albert), frequency assignment problems, and job shop scheduling instances. Across these datasets, SSLD matches or beats DSATUR in almost every case, and outperforms a naive 1-color-class preprocessing baseline referred to as GISD.

The quality improvement comes with a significant runtime trade-off: SSLD is roughly 195 times slower than DSATUR. The authors suggest that this demonstrates the value of SDP-guided preprocessing while highlighting a direction for future work to reduce computational overhead.