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Li and Panigrahi, in recent work, obtained the first deterministic algorithmfor the global minimum cut of a weighted undirected graph that runs in time$o(mn)$. They introduced an elegant and powerful technique to find isolatingcuts for a terminal set in a graph via a small number of $s$-$t$ minimum cutcomputations.  In this paper we generalize their isolating cut approach to the abstractsetting of symmetric bisubmodular functions (which also capture symmetricsubmodular functions). Our generalization to bisubmodularity is motivated byapplications to element connectivity and vertex connectivity. Utilizing thegeneral framework and other ideas we obtain significantly faster randomizedalgorithms for computing global (and subset) connectivity in a number ofsettings including hypergraphs, element connectivity and vertex connectivity ingraphs, and for symmetric submodular functions.

Isolating Cuts, (Bi-)Submodularity, and Faster Algorithms for Global Connectivity Problems