Algorithms for Cut Problems on Trees

TL;DR

This paper introduces improved parameterized algorithms for the multicut on trees problem and shows polynomial-time solvability for the generalized multiway cut on trees when the number of terminal sets is fixed, advancing the understanding of these graph cut problems.

Contribution

The paper presents a faster parameterized algorithm for multicut on trees and resolves the open problem of polynomial-time solvability for fixed terminal sets in generalized multiway cut on trees.

Findings

Improved algorithm with runtime O*(1.555^k) for multicut on trees.

Polynomial-time solvability for fixed number of terminal sets in generalized multiway cut on trees.

Reduction from generalized multiway cut to multicut on trees for algorithmic solution.

Abstract

We study the {\sc multicut on trees} and the {\sc generalized multiway Cut on trees} problems. For the {\sc multicut on trees} problem, we present a parameterized algorithm that runs in time $O^{*}(\rho^k)$, where $\rho = \sqrt{\sqrt{2} + 1} \approx 1.555$ is the positive root of the polynomial $x^4-2x^2-1$. This improves the current-best algorithm of Chen et al. that runs in time $O^{*}(1.619^k)$. For the {\sc generalized multiway cut on trees} problem, we show that this problem is solvable in polynomial time if the number of terminal sets is fixed; this answers an open question posed in a recent paper by Liu and Zhang. By reducing the {\sc generalized multiway cut on trees} problem to the {\sc multicut on trees} problem, our results give a parameterized algorithm that solves the {\sc generalized multiway cut on trees} problem in time $O^{*}(\rho^k)$, where $\rho = \sqrt{\sqrt{2} + 1} \approx 1.555$ time.