@inproceedings{11852,
  abstract     = {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 O~(n2/3) 11The O~(⋅) notation is used in this paper to hide poly-logarithmic factors. amortized time against an oblivious adversary, and O~(m3/4) time against an adaptive adversary. (2)An incremental data structure that maintains O(1) - approximate shortest path in no(1) time per operation, as well as fully dynamic approximate all-pair shortest path and transshipment in O~(n2/3+o(1)) amortized time per operation. (3)A fully-dynamic algorithm that approximates all-pair effective resistance up to an (1+ϵ) factor in O~(n2/3+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. See https://arxiv.org/pdf/2005.02368.pdf for the full version of this paper.},
  author       = {Chen, Li and Goranci, Gramoz and Henzinger, Monika H and Peng, Richard and Saranurak, Thatchaphol},
  booktitle    = {61st Annual Symposium on Foundations of Computer Science},
  isbn         = {978-1-7281-9622-0},
  issn         = {2575-8454},
  location     = {Durham, NC, United States},
  pages        = {1135--1146},
  publisher    = {Institute of Electrical and Electronics Engineers},
  title        = {{Fast dynamic cuts, distances and effective resistances via vertex sparsifiers}},
  doi          = {10.1109/focs46700.2020.00109},
  year         = {2020},
}