Asymptotic analysis of dynamic degree greedy

Analyze the asymptotic performance of the dynamic degree greedy algorithm on sparse Erdős–Rényi graphs by tracking the full empirical degree distribution of the evolving unexplored subgraph G[U_t].

Background

The paper studies dynamic degree greedy (DG), which repeatedly selects a minimum-degree vertex in the subgraph induced by the currently unexplored vertices. Unlike static degree greedy, DG recomputes degrees throughout the exploration process, so its evolution cannot be described by the finite phase recursion used for static degree greedy. A rigorous asymptotic analysis would require tracking the complete empirical degree distribution of the evolving unexplored subgraph; the paper does not carry out this analysis and explicitly leaves it for future work.

References

Thus, analyzing DG asymptotically requires tracking the full empirical degree distribution of $G[\mathcal U_{t}]$, rather than a finite phase recursion, which we leave for future work.

— On the Slow Convergence to Trivial Solutions of Algorithms for Hard Optimization Problems  (2608.18910 - Umar et al., 19 Aug 2026) in Section 3, Discussion of the results for MIS, subsection “Results for sequential greedy algorithms”