Faster Minimum k-cut of a Simple Graph

We consider the (exact, minimum) $k$-cut problem: given a graph and an integer $k$, delete a minimum-weight set of edges so that the remaining graph has at least $k$ connected components. This problem is a natural generalization of the global minimum cut problem, where the goal is to break the graph into $k=2$ pieces. Our main result is a (combinatorial) $k$-cut algorithm on simple graphs that runs in $n^{(1+o(1))k}$ time for any constant $k$, improving upon the previously best $n^{(2\omega/3+o(1))k}$ time algorithm of Gupta et al.~[FOCS'18] and the previously best $n^{(1.981+o(1))k}$ time combinatorial algorithm of Gupta et al.~[STOC'19]. For combinatorial algorithms, this algorithm is optimal up to $o(1)$ factors assuming recent hardness conjectures: we show by a straightforward reduction that $k$-cut on even a simple graph is as hard as $(k-1)$-clique, establishing a lower bound of $n^{(1-o(1))k}$ for $k$-cut. This settles, up to lower-order factors, the complexity of $k$-cut on a simple graph for combinatorial algorithms.