# VMware® Research

### Abstract

We initiate the study of 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 $(1 \pm \epsilon)$-spectral sparsifier with amortized update time $poly(\log{n}, \epsilon^{-1})$. Second, we give a fully dynamic algorithm for maintaining a $(1 \pm \epsilon)$-cut sparsifier with \emph{worst-case} update time $poly(\log{n}, \epsilon^{-1})$. Both sparsifiers have size $n \cdot poly(\log{n}, \epsilon^{-1})$. Third, we apply our dynamic sparsifier algorithm to obtain a fully dynamic algorithm for maintaining a $(1 - \epsilon)$-approximate minimum cut in an unweighted, undirected, bipartite graph with amortized update time $poly(\log{n}, \epsilon^{-1})$.

2016

#### Authors

• Ittai Abraham
• David Durfee
• Ioannis Koutis
• Sebastian Krinninger
• Richard Peng

Conference

FOCS