Multicut in Trees Viewed through the Eyes of Vertex Cover

Chen, Jianer; Fan, Jia-Hao; Kanj, Iyad A.; Liu, Yang; Zhang, Fenghui

doi:10.1007/978-3-642-22300-6_19

Cited by 4 publications

(11 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Chen et al [3] that runs in time O * (1.619 k ). This improvement is obtained by extending the connection between the MCT problem and the Vertex Cover problem, first exploited in the paper of Chen et al [3].…”

Section: Our Resultsmentioning

confidence: 99%

“…All the reduction rules and branching rules in this section appear in [3]. To state these rules as simple and clear as possible, we first make certain some common notations.…”

Section: Reduction Rules For Mctmentioning

confidence: 99%

“…Another interesting research direction is related to kernelization. Currently, MCT is know to admit a kernel of size O (k 3 ) [3], and it is interesting to investigate if the problem admits a quadratic, or even a linear, kernel. The O (k 3 ) kernel for MCT relies heavily on kernelization techniques used for Vertex Cover.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Improved parameterized and exact algorithms for cut problems on trees

Kanj

Lin

Liu

et al. 2015

Theoretical Computer Science

Self Cite

View full text Add to dashboard Cite

Section: Our Resultsmentioning

confidence: 99%

“…All the reduction rules and branching rules in this section appear in [3]. To state these rules as simple and clear as possible, we first make certain some common notations.…”

Section: Reduction Rules For Mctmentioning

confidence: 99%

See 1 more Smart Citation

Improved parameterized and exact algorithms for cut problems on trees

Kanj

Lin

Liu

et al. 2015

Theoretical Computer Science

Self Cite

View full text Add to dashboard Cite

“…They also showed that MCT has an exponential-size kernel. Bousquet, Daligault, Thomassé, and Yeo, improved the upper bound on the kernel size for MCT to O(k 6 ) [2], which was subsequently improved very recently by Chen et al [3] to O(k 3 ). Chen et al[3] also gave a parameterized algorithm for the problem running in time O * (1.619 k ).The multiway cut on trees problem (i.e., there is one set of terminals) was proved to be solvable in polynomial time in [6,8].…”

mentioning

confidence: 99%

mentioning

confidence: 99%

Algorithms for Cut Problems on Trees

Kanj

Lin

Liu

et al. 2014

Combinatorial Optimization and Applications

Self Cite

View full text Add to dashboard Cite

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 * (ρ k ), where ρ = √ 2 + 1 ≈ 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 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 * (ρ k ), where ρ = √ 2 + 1 ≈ 1.555 time.For convenience, we will refer to a pair of terminals (u i , v i ) ∈ R by a request, and we will also say that u i has a request to v i , and vice versa. generalized multiway cut on trees (GMWCT) Given: A tree T and and a collection of vertex/terminal-sets S 1 , . . . S r Parameter: k Question: Is there a set of at most k edges in T whose removal disconnects each pair of vertices in the same terminal set S i , for i = 1, . . . , r?As the name indicates, the GMWCT problem generalizes the well-known multiway cut on trees problem in which there is only one set of terminals.The MCT problem has applications in networking [7]. The problem is known to be NPcomplete, and its optimization version is APX-complete and has an approximation ratio of 2 [11]. Assuming the Unique Games Conjecture, the MCT problem cannot be approximated to within 2 − ǫ [13]. From the parameterized complexity perspective, Guo and Niedermeier [12] showed that the MCT problem is fixed-parameter tractable by giving an O * (2 k ) time algorithm for the problem. (The asymptotic notation O * (f (k)) suppresses any polynomial factor in the input length.) They also showed that MCT has an exponential-size kernel. Bousquet, Daligault, Thomassé, and Yeo, improved the upper bound on the kernel size for MCT to O(k 6 ) [2], which was subsequently improved very recently by Chen et al. [3] to O(k 3 ). Chen et al.[3] also gave a parameterized algorithm for the problem running in time O * (1.619 k ).The multiway cut on trees problem (i.e., there is one set of terminals) was proved to be solvable in polynomial time in [6,8]. Chopra and Rao [6] first gave a polynomial-time greedy algorithm for the problem. More recently, Costa and Billionnet [8] proved that multiway cut on trees can be solved in linear time by dynamic programming. Very recently, Liu and Zhang [15] generalized the multiway cut on trees problem from one set of terminals to allowing multiple terminal sets, which results in the GMWCT defined above. They showed that the GMWCT problem is fixed-parameter tractable by reducing it to the MCT problem [15]. Clearly, the GMWCT problem is NP-complete when the number of terminal sets is part of the input by a trivial reduction from the MCT problem. Liu and Zhan...

show abstract

Parameterized complexity and approximation issues for the colorful components problems

Dondi

Sikora

2018

Theoretical Computer Science

View full text Add to dashboard Cite

The quest for colorful components (connected components where each color is associated with at most one vertex) inside a vertex-colored graph has been widely considered in the last ten years. Here we consider two variants, Minimum Colorful Components (MCC) and Maximum Edges in transitive Closure (MEC), introduced in 2011 in the context of orthology gene identification in bioinformatics. The input of both MCC and MEC is a vertex-colored graph. MCC asks for the removal of a subset of edges, so that the resulting graph is partitioned in the minimum number of colorful connected components; MEC asks for the removal of a subset of edges, so that the resulting graph is partitioned in colorful connected components and the number of edges in the transitive closure of such a graph is maximized. We study the parameterized and approximation complexity of MCC and MEC, for general and restricted instances.For MCC on trees we show that the problem is basically equivalent to Minimum Cut on Trees, thus MCC is not approximable within factor 1.36 − ε, it is fixed-parameter tractable and it admits a poly-kernel (when the parameter is the number of colorful components). Moreover, we show that MCC, while it is polynomial time solvable on paths, it is NP-hard even for graphs with constant distance to disjoint paths number. Then we consider the parameterized complexity of MEC when parameterized by the number k of edges in the transitive closure of a solution (the graph obtained by removing edges so that it is partitioned in colorful connected components). We give a fixed-parameter algorithm for MEC paramterized by k and, when the input graph is a tree, we give a poly-kernel.

show abstract

Multicut in Trees Viewed through the Eyes of Vertex Cover

Cited by 4 publications

References 8 publications

Improved parameterized and exact algorithms for cut problems on trees

Improved parameterized and exact algorithms for cut problems on trees

Algorithms for Cut Problems on Trees

Parameterized complexity and approximation issues for the colorful components problems

Contact Info

Product

Resources

About