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

TL;DR

This paper extends isolating cut techniques to bisubmodular functions, enabling faster randomized algorithms for global connectivity problems in graphs, hypergraphs, and submodular functions.

Contribution

It generalizes the isolating cut approach to bisubmodular functions, leading to improved algorithms for various connectivity problems.

Findings

Faster randomized algorithms for global connectivity.

Extension of isolating cut techniques to bisubmodular functions.

Applications to hypergraphs, element, and vertex connectivity.

Abstract

Li and Panigrahi, in recent work, obtained the first deterministic algorithm for the global minimum cut of a weighted undirected graph that runs in time $o(mn)$. They introduced an elegant and powerful technique to find isolating cuts for a terminal set in a graph via a small number of $s$-$t$ minimum cut computations. In this paper we generalize their isolating cut approach to the abstract setting of symmetric bisubmodular functions (which also capture symmetric submodular functions). Our generalization to bisubmodularity is motivated by applications to element connectivity and vertex connectivity. Utilizing the general framework and other ideas we obtain significantly faster randomized algorithms for computing global (and subset) connectivity in a number of settings including hypergraphs, element connectivity and vertex connectivity in graphs, and for symmetric submodular functions.