Fast Dynamic Cuts, Distances and Effective Resistances via Vertex Sparsifiers

We present a general framework of designing efficient dynamic approximate algorithms for optimization problems on undirected graphs. In particular, we develop a technique that, given any problem that admits a certain notion of vertex sparsifiers, gives data structures that maintain approximate solutions in sub-linear update and query time. We illustrate the applicability of our paradigm to the following problems. (1)A fully-dynamic algorithm that approximates all-pair maximum-flows/minimum-cuts up to a nearly logarithmic factor in
$\tilde{O}(n^{2/3})$

^1
1
The
$\tilde{O}(\cdot)$
notation is used in this paper to hide poly-logarithmic factors. amortized time against an oblivious adversary, and
$\tilde{O}(m^{3/4})$
time against an adaptive adversary. (2)An incremental data structure that maintains
$O(1)$
- approximate shortest path in
$n^{o(1)}$
time per operation, as well as fully dynamic approximate all-pair shortest path and transshipment in
$\tilde{O}(n^{2/3 +o(1)})$
amortized time per operation. (3)A fully-dynamic algorithm that approximates all-pair effective resistance up to an (
$1+\epsilon$
) factor in
$\tilde{O}(n^{2/3+o(1)}\epsilon^{-O(1)})$
amortized update time per operation. The key tool behind result (1) is the dynamic maintenance of an algorithmic construction due to Madry [FOCS' 10], which partitions a graph into a collection of simpler graph structures (known as
$j$
-trees) and approximately captures the cut-flow and metric structure of the graph. The
$O(1)$
-approximation guarantee of (2) is by adapting the distance oracles by [Thorup-Zwick JACM '05]. Result (3) is obtained by invoking the random-walk based spectral vertex sparsifier by [Durfee et al. STOC '19] in a hierarchical manner, while carefully keeping track of the recourse among levels in the hierarchy. See https://arxiv.org/pdf/2005.02368.pdf for the full version of this paper.