Approximation Algorithms for Some Graph Partitioning Problems

He, George; Liu, J.; Zhao, Cheng

doi:10.1142/9789812794741_0002

Cited by 10 publications

(32 citation statements)

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“…In this section, we describe the first approximation algorithm for the maximum-weighted k-partite matching problem. Our approximation algorithm is a generalization of the approximation algorithm for the maximum disjoint kclique problem in [7], which is a special case of maximumweighted k-partite matching problem. The approximation algorithm given in [7] is based on the maximum-weighted matching in a bipartite graph and the idea of merging matched vertices.…”

Section: An Approximation Algorithm For the Maximum-weighted K-pamentioning

confidence: 99%

“…Our approximation algorithm is a generalization of the approximation algorithm for the maximum disjoint kclique problem in [7], which is a special case of maximumweighted k-partite matching problem. The approximation algorithm given in [7] is based on the maximum-weighted matching in a bipartite graph and the idea of merging matched vertices. Our approximation algorithm for the maximum-weighted k-partite matching problem is based on the maximum-weighted (1, r)-matching in a bipartite graph and the idea of merging matched vertices.…”

Section: An Approximation Algorithm For the Maximum-weighted K-pamentioning

confidence: 99%

“…First, we extend the definition of the merging operation in [7] in order to merge (1, r)-matchings. For clarity, we reproduce some notations from [7] in the following definition.…”

Section: An Approximation Algorithm For the Maximum-weighted K-pamentioning

confidence: 99%

“…For clarity, we reproduce some notations from [7] in the following definition. In our definition, the matched vertices are merged into a single node, whereas in [7], an edge is merged into a node. …”

Section: An Approximation Algorithm For the Maximum-weighted K-pamentioning

confidence: 99%

“…We now propose an approximation algorithm for the maximum-weighted k-partite matching problem (see Algorithm 3) that is the generalization of the approximation algorithm for the maximum disjoint k-clique problem in [7]. In order to explain our algorithm clearly, we present our algorithm in a modified format from the description of the approximation algorithm in [7].…”

Section: An Approximation Algorithm For the Maximum-weighted K-pamentioning

confidence: 99%

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The Multiple Alignment Algorithm for Metabolic Pathways without Abstraction

Chen

Rocha

Hendrix

et al. 2010

2010 IEEE International Conference on Data Mining Workshops

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Computational problems associated with metabolic pathways have been extensively studied in computational biology. The problem of aligning multiple metabolic pathways is very challenging. Tohsato et al.'s algorithm for aligning multiple metabolic pathways [22] is based on similarities between enzymes; however, a metabolic pathway consists of three types of entities: reactions, compounds, and enzymes. In this paper, we propose the first algorithm for the problem of aligning multiple metabolic pathways based on the similarities among reactions, compounds, enzymes, and pathway topology.First, we compute a weight between each pair of like entities in different input pathways based on the entities' similarity score and topological structure using the methods in [1]. We then construct a weighted k-partite graph for the reactions, compounds, and enzymes. We extract a mapping between these entities by solving the maximum-weighted kpartite matching problem by applying a novel heuristic algorithm. By analyzing the alignment results of multiple pathways in different organisms, we show that the alignments found by our algorithm correctly identify common subnetworks among multiple pathways.

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Section: An Approximation Algorithm For the Maximum-weighted K-pamentioning

confidence: 99%

Section: An Approximation Algorithm For the Maximum-weighted K-pamentioning

confidence: 99%

“…First, we extend the definition of the merging operation in [7] in order to merge (1, r)-matchings. For clarity, we reproduce some notations from [7] in the following definition.…”

Section: An Approximation Algorithm For the Maximum-weighted K-pamentioning

confidence: 99%

Section: An Approximation Algorithm For the Maximum-weighted K-pamentioning

confidence: 99%

Section: An Approximation Algorithm For the Maximum-weighted K-pamentioning

confidence: 99%

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The Multiple Alignment Algorithm for Metabolic Pathways without Abstraction

Chen

Rocha

Hendrix

et al. 2010

2010 IEEE International Conference on Data Mining Workshops

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Towards a Complexity Dichotomy for Colourful Components Problems on k-caterpillars and Small-Degree Planar Graphs

Chlebíková

Dallard

2019

Lecture Notes in Computer Science

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A connected component of a vertex-coloured graph is said to be colourful if all its vertices have different colours, and a graph is colourful if all its connected components are colourful. Given a vertex-coloured graph, the Colourful Components problem asks whether there exist at most p edges whose removal makes the graph colourful, and the Colourful Partition problem asks whether there exists a partition of the vertex set with at most p parts such that each part induces a colourful component. We study the problems on k-caterpillars (caterpillars with hairs of length at most k) and explore the boundary between polynomial and NP-complete cases. It is known that the problems are NP-complete on 2-caterpillars with unbounded maximum degree. We prove that both problems remain NP-complete on binary 4-caterpillars and on ternary 3caterpillars. This answers an open question regarding the complexity of the problems on trees with maximum degree at most 5. On the positive side, we give a linear time algorithm for 1-caterpillars with unbounded degree, even if the backbone is a cycle, which outperforms the previous best complexity for paths and widens the class of graphs. Finally, we answer an open question regarding the complexity of Colourful Components on graphs with maximum degree at most 5. We show that the problem is NP-complete on 5-coloured planar graphs with maximum degree 4, and on 12-coloured planar graphs with maximum degree 3. Since the problem can be solved in polynomial-time on graphs with maximum degree 2, the results are the best possible with regard to the maximum degree.

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