Efficient Algorithms for a Graph Partitioning Problem
Motivated by an expensive computation performed by a computational topology software RIVET [9], Madkour et al. [1] introduced and studied the following graph partitioning problem. Given an edge weighted graph and an integer k, partition the vertex set of the graph into k connected components such th...
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| Published in | Frontiers in Algorithmics Vol. 10823; pp. 29 - 42 |
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| Main Authors | , , |
| Format | Book Chapter |
| Language | English |
| Published |
Switzerland
Springer International Publishing AG
2018
Springer International Publishing |
| Series | Lecture Notes in Computer Science |
| Online Access | Get full text |
| ISBN | 3319784544 9783319784540 |
| ISSN | 0302-9743 1611-3349 |
| DOI | 10.1007/978-3-319-78455-7_3 |
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| Summary: | Motivated by an expensive computation performed by a computational topology software RIVET [9], Madkour et al. [1] introduced and studied the following graph partitioning problem. Given an edge weighted graph and an integer k, partition the vertex set of the graph into k connected components such that the weight of the heaviest component is as small as possible, where the weight of each component is the weight of a minimum spanning tree of the graph induced by the vertices in that component. They showed that the problem is NP-hard even for $$k=2$$ and provided some heuristic algorithms. They asked whether the problem is polynomial time solvable when the input is a tree. Our first result is an affirmative answer to their question. We give a polynomial time algorithm to provide such a partition in a tree. We also give an exact exponential algorithm taking $$O^*(2^n)$$ time on general graphs (improving on the naive $$O^*(k^n)$$ algorithm) ( $$O^*$$ notation ignores polynomial factors). We also prove that the problem remains NP-complete even when the weights on all the edges are the same and give a linear time algorithm for this version of the problem when the graph is a tree. |
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| Bibliography: | Original Abstract: Motivated by an expensive computation performed by a computational topology software RIVET [9], Madkour et al. [1] introduced and studied the following graph partitioning problem. Given an edge weighted graph and an integer k, partition the vertex set of the graph into k connected components such that the weight of the heaviest component is as small as possible, where the weight of each component is the weight of a minimum spanning tree of the graph induced by the vertices in that component. They showed that the problem is NP-hard even for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=2$$\end{document} and provided some heuristic algorithms. They asked whether the problem is polynomial time solvable when the input is a tree. Our first result is an affirmative answer to their question. We give a polynomial time algorithm to provide such a partition in a tree. We also give an exact exponential algorithm taking \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O^*(2^n)$$\end{document} time on general graphs (improving on the naive \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O^*(k^n)$$\end{document} algorithm) (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O^*$$\end{document} notation ignores polynomial factors). We also prove that the problem remains NP-complete even when the weights on all the edges are the same and give a linear time algorithm for this version of the problem when the graph is a tree. Work done while the authors were visiting IMSc Chennai. |
| ISBN: | 3319784544 9783319784540 |
| ISSN: | 0302-9743 1611-3349 |
| DOI: | 10.1007/978-3-319-78455-7_3 |