Fast Dynamic Cuts, Distances and Effective Resistances via Vertex Sparsifiers
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We present a general framework of designing efficient dynamic approximate algorithms for optimization 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 Õ(n^2/3) amortized time against an oblivious adversary, and Õ(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 Õ(n^2/3+o(1)) amortized time per operation. (3) A fully-dynamic algorithm that approximates all-pair effective resistance up to an (1+ϵ) factor in Õ(n^2/3+o(1)ϵ^-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.
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