SSLD Fixes One SDP Color Class to Match or Beat DSATUR on 1600+ Instances

The method extracts the first color class from an SDP relaxation close to the Lovász theta formulation, then hands the residual graph to DSATUR. This single fixed class replaces the greedy first step that normally produces excess colors. Runtime rises by a factor of roughly 195, yet the preprocessing step itself is a one-time SDP solve whose output is reused across multiple DSATUR runs on the same graph.

On the full test set SSLD never increases the number of colors and improves it on the majority of instances; the gap widens on structured graphs from scheduling and frequency-assignment benchmarks where DSATUR’s saturation ordering alone leaves many vertices poorly ordered. The naive GISD baseline, which selects a maximum independent set without SDP guidance, is consistently outperformed, confirming that the spectral choice of the seed class carries measurable value beyond mere size.

The approach directly lowers the effective branching factor for downstream exact solvers that invoke DSATUR as a primal heuristic. In network design and timetabling pipelines that already tolerate multi-minute preprocessing, the added SDP cost fits inside existing nightly batch windows while trimming color counts that translate into fewer channels or fewer time slots.

Future work can amortize the SDP cost across graph families or replace the Lovász-style relaxation with cheaper spectral proxies, opening a route to hybrid exact-heuristic pipelines that remain practical on industrial instances within the next twelve months.