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# Classification of internal vertices.

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Given a tree T = (V, E) with n vertices and a collection of terminal sets D = {S
1, S
2, …, S
c
}, where each S
i
is a subset of V and c is a constant, the generalized Multiway Cut in trees problem (GMWC(T)) asks to find a minimum size edge subset E′ ⊆ E such that its removal from the tree separates all terminals in S
i
from each other for each ter...

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## Citations

... Zhang et al. (2012) introduced the generalized multicut problem in trees and presented an approximation algorithm. Liu and Zhang (2014) introduced the generalized multiway cut problem in trees and showed that it is fixed parameter tractable according to the optimal value, which is improved by Kanj et al. (2015). Levin and Segev (2006) proposed the prize-collecting multicut problem in trees, which is a generalization of the multicut problem in trees. ...

In this paper, we introduce the submodular multicut problem in trees with submodular penalties, which generalizes the prize-collecting multicut problem in trees and the submodular vertex cover with submodular penalties. We present a combinatorial approximation algorithm, based on the primal-dual algorithm for the submodular set cover problem. In addition, we present a combinatorial 3-approximation algorithm for a special case where the edge cost is a modular function, based on the primal-dual scheme for the multicut problem in trees.

... More recently, Costa and Billionnet [7] proved that Multiway Cut on Trees can be solved in linear time by dynamic programming. Very recently, Liu and Zhang [19] 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 fixedparameter tractable by reducing it to the MCT problem [19]. ...

... Very recently, Liu and Zhang [19] 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 fixedparameter tractable by reducing it to the MCT problem [19]. Clearly, the GMWCT problem is NP-complete when the number of terminal sets is part of the input, since it is a generalization of the MCT problem. ...

... Clearly, the GMWCT problem is NP-complete when the number of terminal sets is part of the input, since it is a generalization of the MCT problem. Liu and Zhang asked about the complexity of the problem if the number of terminal sets is a constant (i.e., not part of the input) [19]. ...

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 , where is the positive root of the polynomial . This improves the current-best algorithm of Chen et al. that runs in time . 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 .

... 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]. ...

... 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. ...

... 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 Zhang asked about the complexity of the problem if the number of terminal sets is a constant (i.e., not part of the input) [15]. ...

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.

In this paper, we introduce the multicut problem in trees with submodular penalties, which generalizes the prize-collecting multicut problem in trees and vertex cover with submodular penalties. We present a combinatorial 3-approximation algorithm, based on the primal-dual scheme for the multicut problem in trees.

Microdiesel obtained from microbes using renewable materials as carbon sources is an important alternative to petroleum diesel. This review provides information related to microdiesel production using various carbon sources; i.e. carbon dioxide, C2, saccharides, and lignocellulose. Microbes can accumulate different contents of fatty acids in the form of triacylglycerol (TAG). Not all microbes store fatty acids and utilize a broad range of substrates as carbon sources, and vice versa. Microbes can be engineered to consume various carbon sources, and accumulate increased amounts of fatty acids with different composition. The properties of microdiesel depend on its fatty acid profile, which in turn determines its efficacy. The structural features of the fatty acids, such as carbon chain length, branching and degree of unsaturation, affect the physiochemical properties of the biodiesel (cetane number (CN), oxidation stability (OS), iodine value (IV), cold flow properties, density and kinematic viscosity). Fatty acid methyl ester (FAME) profiles can be used to evaluate the key properties of biodiesel, i.e. the stability of the oil used. The overview presented herein concludes that microdiesel production using non-feed carbon sources and genetically engineered microbes shows much promise.

The Multi-Multiway Cut problem proposed by Avidor and Langberg [2] is a natural generalization of Multicut and Multiway Cut problems. That is, given a simple graph G and c sets of vertices S
1, ⋯ ,S
c
, the problem asks for a minimum set of edges whose removal disconnects every pair of vertices in S
i
for all 1 ≤ i ≤ c. In [13], the authors asked whether the problem is polynomial time solvable for fixed c on trees. In this paper, we give both a logical approach and a dynamic programming approach to the Multi-Multiway Cut problem on graphs of bounded branch width, which is exactly the class of graphs with bounded treewidth. In fact, for fixed c and branch width k, we show that the Multi-Multiway Cut problem can be solved in linear time, thus affirmatively answer the question in [13].