Sub-$n^k$ Deterministic algorithm for minimum $k$-way cut in simple graphs

Abstract

We present a \emph{deterministic exact algorithm} for the \emph{minimum $k$-cut problem} on simple graphs. Our approach combines the \emph{principal sequence of partitions (PSP)}, derived canonically from ideal loads, with a single level of \emph{Kawarabayashi--Thorup (KT)} contractions at the critical PSP threshold~$\lambda_j$. Let $j$ be the smallest index with $\kappa(P_j)\ge k$ and $R := k - \kappa(P_{j-1})$. We prove a structural decomposition theorem showing that an optimal $k$-cut can be expressed as the level-$(j\!-\!1)$ boundary $A_{\le j-1}$ together with exactly $(R-r)$ \emph{non-trivial} internal cuts of value at most~$\lambda_j$ and $r$ \emph{singleton isolations} (``islands'') inside the parts of~$P_{j-1}$. At this level, KT contractions yield kernels of total size $\widetilde{O}(n / \lambda_j)$, and from them we build a \emph{canonical border family}~$\mathcal{B}$ of the same order that deterministically covers all optimal refinement choices. Branching only over~$\mathcal{B}$ (and also including an explicit ``island'' branch) gives total running time $$ T(n,m,k) = \widetilde{O}\left(\mathrm{poly}(m)+\Bigl(\tfrac{n}{\lambda_j}+n^{\omega/3}\Bigr)^{R}\right), $$ where $\omega < 2.373$ is the matrix multiplication exponent. In particular, if $\lambda_j \ge n^{\varepsilon}$ for some constant $\varepsilon > 0$, we obtain a \emph{deterministic sub-$n^k$-time algorithm}, running in $n^{(1-\varepsilon)(k-1)+o(k)}$ time. Finally, combining our PSP$\times$KT framework with a small-$\lambda$ exact subroutine via a simple meta-reduction yields a deterministic $n^{c k+O(1)}$ algorithm for $c = \max\{ t/(t+1), \omega/3 \} < 1$, aligning with the exponent in the randomized bound of He--Li (STOC~2022) under the assumed subroutine.