Minimum Cut and Minimum k-Cut in Hypergraphs via Branching Contractions

Abstract

On hypergraphs with m hyperedges and n vertices, where p denotes the total size of the hyperedges, we provide the following results:

•	We give an algorithm that runs in \(\widetilde{O}\left(mn^{2k-2}\right) \) time for finding a minimum k-cut in hypergraphs of arbitrary rank. This algorithm betters the previous best running time for the minimum k-cut problem, for k > 2.
•	We give an algorithm that runs in \(\widetilde{O}\left(n^{\max \lbrace r,2k-2\rbrace }\right) \) time for finding a minimum k-cut in hypergraphs of constant rank r. This algorithm betters the previous best running times for both the minimum cut and minimum k-cut problems for dense hypergraphs.

Both of our algorithms are Monte Carlo, i.e., they return a minimum k-cut (or minimum cut) with high probability. These algorithms are obtained as instantiations of a generic branching randomized contraction technique on hypergraphs, which extends the celebrated work of Karger and Stein on recursive contractions in graphs. Our techniques and results also extend to the problems of minimum hedge-cut and minimum hedge-k-cut on hedgegraphs, which generalize hypergraphs.