An $O^*(1.84^k)$ Parameterized Algorithm for the Multiterminal Cut Problem

We study the \emph{multiterminal cut} problem, which, given an $n$-vertex graph whose edges are integer-weighted and a set of terminals, asks for a partition of the vertex set such that each terminal is in a distinct part, and the total weight of crossing edges is at most $k$. Our weapons shall be two classical results known for decades: \emph{maximum volume minimum ($s,t$)-cuts} by [Ford and Fulkerson, \emph{Flows in Networks}, 1962] and \emph{isolating cuts} by [Dahlhaus et al., \emph{SIAM J. Comp.} 23(4):864-894, 1994]. We sharpen these old weapons with the help of submodular functions, and apply them to this problem, which enable us to design a more elaborated branching scheme on deciding whether a non-terminal vertex is with a terminal or not. This bounded search tree algorithm can be shown to run in $1.84^k\cdot n^{O(1)}$ time, thereby breaking the $2^k\cdot n^{O(1)}$ barrier. As a by-product, it gives a $1.36^k\cdot n^{O(1)}$ time algorithm for $3$-terminal cut. The preprocessing applied on non-terminal vertices might be of use for study of this problem from other aspects.