Algorithms for Cut Problems on Trees

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

Parameterized Complexity of Weighted Multicut in Trees

The Edge Multicut problem is a classical cut problem where given anundirected graph $G$, a set of pairs of vertices $\mathcal{P}$, and a budget$k$, the goal is to determine if there is a set $S$ of at most $k$ edges suchthat for each $(s,t) \in \mathcal{P}$, $G-S$ has no path from $s$ to $t$. EdgeMulticut has been relatively recently shown to be fixed-parameter tractable(FPT), parameterized by $k$, by Marx and Razgon [SICOMP 2014], andindependently by Bousquet et al. [SICOMP 2018]. In the weighted version of theproblem, called Weighted Edge Multicut one is additionally given a weightfunction $\mathtt{wt} : E(G) \to \mathbb{N}$ and a weight bound $w$, and thegoal is to determine if there is a solution of size at most $k$ and weight atmost $w$. Both the FPT algorithms for Edge Multicut by Marx et al. and Bousquetet al. fail to generalize to the weighted setting. In fact, the weightedproblem is non-trivial even on trees and determining whether Weighted EdgeMulticut on trees is FPT was explicitly posed as an open problem by Bousquet etal. [STACS 2009]. In this article, we answer this question positively bydesigning an algorithm which uses a very recent result by Kim et al. [STOC2022] about directed flow augmentation as subroutine. We also study a variant of this problem where there is no bound on the sizeof the solution, but the parameter is a structural property of the input, forexample, the number of leaves of the tree. We strengthen our results by statingthem for the more general vertex deletion version.<br

Approximation algorithms for multi-multiway cut and multicut problems on directed graphs

In this paper, we study the directed multicut and directed multimultiway cut problems. The input to the directed multi-multiway cut problem is a weighted directed graph $G=(V,E)$ and $k$ sets $S_1, S_2,\cdots, S_k$ of vertices. The goal is to find a subset of edges of minimum total weight whose removal will disconnect all the connections between the vertices in each set $S_i$, for $1\leq i\leq k$. A special case of this problem is the directed multicut problem whose input consists of a weighted directed graph $G=(V,E)$ and a set of ordered pairs of vertices $(s_1,t_1),\cdots,(s_k,t_k)$. The goal is to find a subset of edges of minimum total weight whose removal will make for any $i, 1\leq i\leq k$, there is no directed path from si to ti . In this paper, we present two approximation algorithms for these problems. The so called region growing paradigm is modified and used for these two cut problems on directed graphs. using this paradigm, we give an approximation algorithm for each problem such that both algorithms have the approximation factor of $O(k)$ the same as the previous works done on these problems. However, the previous works need to solve $k$ linear programming, whereas our algorithms require only one linear programming. Therefore, our algorithms improve the running time of the previous algorithms

Parameterized Complexity of Weighted Multicut in Trees

The \textsc{Edge Multicut} problem is a classical cut problem where given an undirected graph $G$, a set of pairs of vertices $\mathcal{P}$, and a budget $k$, the goal is to determine if there is a set $S$ of at most $k$ edges such that for each $(s,t) \in \mathcal{P}$, $G-S$ has no path from $s$ to $t$. \textsc{Edge Multicut} has been relatively recently shown to be fixed-parameter tractable (FPT), parameterized by $k$, by Marx and Razgon [SICOMP 2014], and independently by Bousquet et al. [SICOMP 2018]. In the weighted version of the problem, called \textsc{Weighted Edge Multicut} one is additionally given a weight function $\texttt{wt} : E(G) \to \mathbb{N}$ and a weight bound $w$, and the goal is to determine if there is a solution of size at most $k$ and weight at most $w$. Both the FPT algorithms for \textsc{Edge Multicut} by Marx et al.\ and Bousquet et al.\ fail to generalize to the weighted setting. In fact, the weighted problem is non-trivial even on trees and determining whether \textsc{Weighted Edge Multicut} on trees is FPT was explicitly posed as an open problem by Bousquet et al.\ [STACS 2009]. In this article, we answer this question positively by designing an algorithm which uses a very recent result by Kim et al.\ [STOC 2022] about directed flow augmentation as subroutine. We also study a variant of this problem where there is no bound on the size of the solution, but the parameter is a structural property of the input, for example, the number of leaves of the tree. We strengthen our results by stating them for the more general vertex deletion version

Improved Parameterized and Exact Algorithms for Cut Problems on Trees

We study the Multicut on Trees and the Generalized Multiway Cut on Trees problems. For the Multicut on Trees problem, we present a parameterized algorithm that runs in time O ∗(ρk), where ρ = √√2 + 1 \u3c 1.554 is the positive root of the polynomial x4 − 2x2 − 1. This improves the current-best algorithm of Chen et al. that runs in time O ∗(1.619k). For the 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 Generalized Multiway Cut on Trees problem to the Multicut on Trees problem, our results give a parameterized algorithm that solves the Generalized Multiway Cut on Trees problem in time O ∗(ρk)