Simple dynamic algorithms for Maximal Independent Set and other problems.

We study three fundamental graph problems in the dynamic setting, namely, Maximal Independent Set (MIS), Maximum Matching and Maximum Flows. We report surprisingly simple and efficient algorithms for them in different dynamic settings. For MIS we improve the state of the art upper bounds, whereas for incremental Maximum Matching and incremental unit capacity Maximum Flow and Maximum Matching, we match the state of the art lower bounds. Recently, Assadi et al. [STOC18] showed that fully dynamic MIS can be maintained in $O(\min\{\Delta,m^{3/4}\})$ amortized update time. We improve this bound to $O(\min\{\Delta,m^{2/3}\})$. Under incremental setting, we further improve this bound to $O(\min\{\Delta,\sqrt{m}\})$. Also, we show that a simple algorithm can maintain MIS optimally under fully dynamic vertex updates and decremental edge updates. Further, Assadi et al. [STOC18] reported hardness in achieving $o(n)$ worst case update complexity for dynamic MIS. We circumvent the problem by proposing a model for dynamic MIS which does not maintain the MIS explicitly, rather allows queries on whether a vertex belongs to some MIS of the graph. In this model we prove that fully dynamic MIS can be maintained in worst case $O(\min\{\Delta,\sqrt{m}\})$ update and query time. Finally, similar to Assadi et al. [STOC18], all our algorithms can be extended to the distributed setting with update complexity of $O(1)$ rounds and adjustments. Dahlgaard [ICALP16] presented lower bounds of amortized $\Omega(n)$ update time for maintaining incremental unweighted Maximum Flow and incremental Maximum Cardinality Matching. We report trivial extensions of two classical algorithms, namely incremental reachability and blossoms algorithm, which match these lower bounds. For completeness, we also report folklore algorithms for these problems in the fully dynamic setting requiring $O(m)$ worst case update time.