Faster Minimum k-Cut II: Near-Optimal and Deterministic for Weighted Graphs

Abstract

The Minimum $k$-Cut problem asks for a minimum-weight set of edges whose removal leaves an undirected weighted graph with at least $k$ connected components. We consider only $k\ge 3$. Under the Max-Weight Clique conjecture, weighted Minimum $k$-Cut requires $n^{k-1-o(1)}$ time for every fixed $k$. The fastest previous algorithm for weighted graphs ran in $n^{k-2}(m+n)(\log n{)}^{O(k^2)}$ randomized time~\cite{LV26}; for $k=3$, this gave an $\tilde{O}(nm)$-time algorithm. We give randomized and deterministic algorithms matching the conditional lower bound in the exponent. On an $n$-vertex, $m$-edge weighted graph, our randomized algorithm runs with high probability in \begin{equation*} k^{O(k^2)}n^{k-1}\log^2n \end{equation*} time. Our deterministic algorithm runs in \begin{equation*} k^{O(k^2)}n^{k-1}\log^{O(1)}n \end{equation*} time. In particular, weighted Minimum $3$-Cut can be solved in $O(n^2{\log }^{2}n)$ randomized time and in $\tilde{O}(n^2)$ deterministic time. The algorithms have two main components. First, we give a faster algorithm for weighted Minimum $3$-Cut. After handling optima with a very small side and optima with two light sides, the remaining optimum has a unique structured side. Tree packing reduces its completion to a batched collection of $2$-respecting cut problems. Second, we reduce Minimum $k$-Cut to Minimum $3$-Cut by enumerating a bounded family of light-cut candidates and recursively completing either side of each candidate. If the enumeration produces too many cuts, then we can instead produce an optimum $k$-cut directly. We derandomize the $3$-cut algorithm using a deterministic near-minimum-cut skeleton, and derandomize the reduction using a specialized $4$-cut algorithm using the skeleton, the constructive light-cut bounds, and the deterministic spectral sparsifier of \cite{BSS12}.