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 $poly(\log{n}, \epsilon^{-1})$.