@online{Abrahamdkkp16,
TITLE = {On Fully Dynamic Graph Sparsifiers},
AUTHOR = {Abraham, Ittai and Durfee, David and Koutis, Ioannis and Krinninger, Sebastian and Peng, Richard},
LANGUAGE = {eng},
URL = {http://arxiv.org/abs/1604.02094},
EPRINT = {1604.02094},
EPRINTTYPE = {arXiv},
YEAR = {2016},
MARGINALMARK = {$\bullet$},
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 $(1 + \epsilon)$-approximation to the value of the maximum flow in an unweighted, undirected, bipartite graph with amortized update time $poly(\log{n}, \epsilon^{-1})$.},
}