A Deterministic Algorithm for Finding All Minimum k-Way Cuts

Let $G=(V,E)$ be an edge-weighted undirected graph with $n$ vertices and $m$ edges. We present a deterministic algorithm to compute a minimum $k$-way cut of $G$ for a given $k$. Our algorithm is a divide-and-conquer method based on a procedure that reduces an instance of the minimum $k$-way cut problem to $O(n^{2k-5})$ instances of the minimum $(\lfloor (k+\sqrt{k})/2\rfloor+1)$-way cut problem, and can be implemented to run in $O(n^{4k/(1-1.71/\sqrt{k}) -31} )$ time. With a slight modification, the algorithm can find all minimum $k$-way cuts in $O(n^{4k/(1-1.71/\sqrt{k}) -16} )$ time.