# Min-cut

## Faster Connectivity in Low-Rank Hypergraphs via Expander Decomposition

We design an algorithm for computing connectivity in hypergraphs which runs in time $\hat O_r(p + \min{\lambda n^2, n^r/\lambda})$, where $p$ is the size, $n$ is the number of vertices, $r$ is the rank (size of the largest hyperedge), and $\lambda$ is the connectivity of the hypergraph. The $\hat O_r(\cdot)$ hides terms that are subpolynomial in the main parameter and terms that depend only on $r$. Our algorithm is faster than existing algorithms when the connectivity $\lambda$ is $\Omega(n^{(r-2)/2})$.

## Work-Optimal Parallel Minimum Cuts for Non-Sparse Graphs

We present the first work-optimal polylogarithmic-depth parallel algorithm for the minimum cut problem on non-sparse graphs. For $m\geq n^{1+\epsilon}$ for any constant $\epsilon>0$, our algorithm requires $O(m \log n)$ work and $O(\log^3 n)$ depth and succeeds with high probability. Its work matches the best $O(m \log n)$ runtime for sequential algorithms [MN STOC’20; GMW SOSA’21]. This improves the previous best work by Geissmann and Gianinazzi [SPAA’18] by $O(\log^3 n)$ factor, while matching the depth of their algorithm.