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

TL;DR

This paper introduces a faster fixed-parameter algorithm for the multiterminal cut problem, improving the runtime from exponential in 2^k to approximately 1.84^k, using classical network flow results and submodular functions.

Contribution

It presents a novel $O^*(1.84^k)$ parameterized algorithm for multiterminal cut, breaking the previous $2^k$ barrier, and applies classical flow results with submodular functions for improved branching.

Findings

Achieves $1.84^k imes n^{O(1)}$ runtime for multiterminal cut

Provides a $1.36^k imes n^{O(1)}$ algorithm for 3-terminal cut

Introduces preprocessing techniques for non-terminal vertices

Abstract

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.