On Fully Dynamic Graph Sparsifiers

TL;DR

This paper introduces the first fully dynamic algorithms for graph sparsification, enabling efficient updates for spectral and cut sparsifiers, and applies these to maintain approximate maximum flow values in dynamic bipartite graphs.

Contribution

It presents novel fully dynamic algorithms for spectral and cut sparsifiers with polylogarithmic update times, and uses these to approximate maximum flow in dynamic bipartite graphs.

Findings

Spectral sparsifiers maintained with polylogarithmic amortized time.

Cut sparsifiers maintained with polylogarithmic worst-case time.

Approximate maximum flow maintained efficiently in dynamic bipartite graphs.

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