On Fully Dynamic Graph Sparsifiers

We initiate the study of fast dynamic algorithms for graph sparsification problems and obtain fully dynamic algorithms, allowing both edge insertions and edge deletions, that take polylogarithmic time after each update in the graph. Our three main results are as follows. First, we give a fully dynamic algorithm for maintaining a (1 ± ϵ)-spectral sparsifier with amortized update time poly(log n, ϵ ^-1 ). Second, we give a fully dynamic algorithm for maintaining a (1 ± ϵ)-cut sparsifier with worst-case update time poly(log n, ϵ ^-1 ). Both sparsifiers have size n · poly(log n, ϵ ^-1 ). Third, we apply our dynamic sparsifier algorithm to obtain a fully dynamic algorithm for maintaining a (1 - ϵ)-approximation to the value of the maximum flow in an unweighted, undirected, bipartite graph with amortized update time poly(log n, ϵ ^-1 ).