In the hypergraph k-cut problem, the input is a hypergraph, and the goal is to find a smallest subset of hyperedges whose removal ensures that the remaining hypergraph has at least k connected components. This problem is known to be at least as hard as the densest k-subgraph problem when k is part of the input (Chekuri-Li, 2015). We present a randomized polynomial time algorithm to solve the hypergraph k-cut problem for constant k. Our algorithm solves the more general hedge k-cut problem when the subgraph induced by every hedge has a constant number of connected components. In the hedge k-cut problem, the input is a hedgegraph specified by a vertex set and a disjoint set of hedges, where each hedge is a subset of edges defined over the vertices. The goal is to find a smallest subset of hedges whose removal ensures that the number of connected components in the remaining underlying (multi-)graph is at least k. Our algorithm is based on random contractions akin to Karger's min cut algorithm. Our main technical contribution is a distribution over the hedges (hyperedges) so that random contraction of hedges (hyperedges) chosen from the distribution succeeds in returning an optimum solution with large probability.
Automated approaches for detecting dependencies in structures created with Computer Aided Design software are critical for developing robust solvers and providing informative user feedback. We model a set of geometric constraints with a bi-colored multigraph and give a graph-based pebble game algorithm that allows us to determine combinatorially if there are generic dependencies. We further use the pebble game to yield a decomposition of the graph into factor graphs which may be used to give a user detailed feedback about dependent substructures in a specific realization of a system of CAD constraints with non-generic properties.