Simple dynamic algorithms for Maximal Independent Set and other problems

Most graphs in real life keep changing with time. These changes can be in the form of insertion or deletion of edges or vertices. Such rapidly changing graphs motivate us to study dynamic graph algorithms. 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 all these problems in different dynamic settings. For dynamic 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{∆,m3/4}) amortized time per update. We improve this bound to O(min{∆,m2/3}) amortized time per update. Under incremental setting, we further improve this bound to O(min{∆,√m}) amortized time per update. 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 the 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{∆,√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 Ω(n) update time for maintaining incremental unit capacity Maximum Flow and incremental Maximum Cardinality Matching in bipartite graphs (and hence in general graphs). We report trivial extensions of two classical algorithms, namely incremental reachability algorithm and blossoms algorithm, which match these lower bounds. Finally, for the sake of completeness, we also report the folklore algorithms for these problems in the fully dynamic setting requiring O(m) worst case time per update. ∗This research work was supported by the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement no. 340506. ar X iv :1 80 4. 01 82 3v 1 [ cs .D S] 5 A pr 2 01 8