Fully Dynamic Effective Resistances

In this paper we consider the \emph{fully-dynamic} All-Pairs Effective Resistance problem, where the goal is to maintain effective resistances on a graph $G$ among any pair of query vertices under an intermixed sequence of edge insertions and deletions in $G$. The effective resistance between a pair of vertices is a physics-motivated quantity that encapsulates both the congestion and the dilation of a flow. It is directly related to random walks, and it has been instrumental in the recent works for designing fast algorithms for combinatorial optimization problems, graph sparsification, and network science. We give a data-structure that maintains $(1+\epsilon)$-approximations to all-pair effective resistances of a fully-dynamic unweighted, undirected multi-graph $G$ with $\tilde{O}(m^{4/5}\epsilon^{-4})$ expected amortized update and query time, against an oblivious adversary. Key to our result is the maintenance of a dynamic \emph{Schur complement}~(also known as vertex resistance sparsifier) onto a set of terminal vertices of our choice. This maintenance is obtained (1) by interpreting the Schur complement as a sum of random walks and (2) by randomly picking the vertex subset into which the sparsifier is constructed. We can then show that each update in the graph affects a small number of such walks, which in turn leads to our sub-linear update time. We believe that this local representation of vertex sparsifiers may be of independent interest

On Fully Dynamic Graph Sparsifiers

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