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

Mohit Daga

Paper

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

-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~

. Let

be the smallest index with

and

. We prove a structural decomposition theorem showing that an optimal

-cut can be expressed as the level-

boundary

together with exactly

\emph{non-trivial} internal cuts of value at most~

and

\emph{singleton isolations} (``islands'') inside the parts of~

. At this level, KT contractions yield kernels of total size

, and from them we build a \emph{canonical border family}~

of the same order that deterministically covers all optimal refinement choices. Branching only over~

(and also including an explicit ``island'' branch) gives total running time

where

is the matrix multiplication exponent. In particular, if

for some constant

, we obtain a \emph{deterministic sub-

-time algorithm}, running in

time. Finally, combining our PSP

KT framework with a small-

exact subroutine via a simple meta-reduction yields a deterministic

algorithm for

, aligning with the exponent in the randomized bound of He--Li (STOC~2022) under the assumed subroutine.