On Fully Dynamic Graph Sparsifiers
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{worstcase} 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 $poly(\log{n}, \epsilon^{1})$.
 Publication:

arXiv eprints
 Pub Date:
 April 2016
 arXiv:
 arXiv:1604.02094
 Bibcode:
 2016arXiv160402094A
 Keywords:

 Computer Science  Data Structures and Algorithms
 EPrint:
 A preliminary version of this paper appears in the 57th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2016)