A Minimum Linear Arrangement Algorithm for Undirected Trees

Shiloach, Yossi

doi:10.1137/0208002

Cited by 88 publications

(85 citation statements)

References 3 publications

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“…This problem is quite similar to the Minimum Linear Arrangement [34] problem on trees. Recall that, a linear arrangement of a graph (V, E) is a mapping π : V → [n].…”

Section: An Exact Algorithm For Minimizing Average Tree-edge Distortionmentioning

confidence: 91%

Approximation Algorithms for Minimizing Average Distortion

2005

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This paper considers embeddings f of arbitrary finite metrics into the line metric so that none of the distances is shrunk by the embedding f ; the quantity of interest is the factor by which the average distance in the metric is stretched. We call this quantity the average distortion of the non-contracting map f .We prove that finding the best embedding of even a tree metric into a line metric so as to minimize the average distortion is NP-hard, and hence focus on approximating the average distortion of the best possible embedding for the given input metric. We give a constant-factor approximation for the problem of embedding general metrics into the line metric. For the case of n-point tree metrics, we provide a quasi-polynomial time approximation scheme (QPTAS) which outputs an embedding with distortion at most (1 + ) times the optimum in time n O(log n/ 2 ) . Finally, when the average distortion is measured only over the endpoints of the edges of an input tree metric, we show how to exploit the structure of tree metrics to give an exact solution in polynomial time.

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“…This problem is quite similar to the Minimum Linear Arrangement [34] problem on trees. Recall that, a linear arrangement of a graph (V, E) is a mapping π : V → [n].…”

Section: An Exact Algorithm For Minimizing Average Tree-edge Distortionmentioning

confidence: 91%

Approximation Algorithms for Minimizing Average Distortion

2005

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“…Yet it is possible that MCA problem might be tenable if restricted to certain special kinds of graphs that have practical significance. Literature has many such instances of polynomial time algorithms for OLA problem, e.g., unweighted trees [25,8], outer planar graphs [10], wheels, complete bipartite graphs [16], etc. Can one hope for the same in case of MCA?…”

Section: Discussionmentioning

confidence: 99%

“…Garey, Johnson and Stockmeyer [14] proved NP-completeness of the decision version of OLA. Today we know how to solve OLA problem exactly for some special cases of graphs, e.g., un-weighted trees [25,8], outer planar graphs [10], cycles, wheels, complete bipartite graphs [16], etc. For arbitrary graphs, the currently best known guarantee of O(log n)-approximation is due to Rao and Richa [23].…”

Section: Related Problemsmentioning

confidence: 99%

On Minimum Circular Arrangement

Ganapathy

Lodha²

2004

Lecture Notes in Computer Science

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Abstract. Motivated by a scheduling problem encountered in multicast environments, we study a vertex labelling problem, called Minimum Circular Arrangement (MCA), that requires one to find an embedding of a given weighted directed graph into a discrete circle which minimizes the total weighted arc length. Its decision version is already known to be NP-complete when restricted to sparse weighted instances. We prove that the decision version of even un-weighted MCA is NP-complete in case of sparse as well as dense graphs.We also consider complementary version of MCA, called MaxCA. We prove that it is MAX-SNP[π] complete and, therefore, has no PTAS unless P=NP. A similar proof technique shows that MCA is MAX-SNP[π]-Hard and hence admits no PTAS as well. Then we prove a conditional lower bound of √ 2 − for MCA approximation under some hardness assumptions, and conclude with a PTAS for MCA on dense instances.

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“…The problem is solvable in polynomial time for trees [6,3,18], and for some other restricted graph classes such as grids or hypercubes [4]. There is an approximation algorithm for general graphs with performance ratio O(log n) [17].…”

Section: Introductionmentioning

confidence: 99%

Optimal Linear Arrangement of Interval Graphs

Cohen

Fomin

Heggernes

et al. 2006

Lecture Notes in Computer Science

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Abstract. We study the optimal linear arrangement (OLA) problem on interval graphs. Several linear layout problems that are NP-hard on general graphs are solvable in polynomial time on interval graphs. We prove that, quite surprisingly, optimal linear arrangement of interval graphs is NP-hard. The same result holds for permutation graphs. We present a lower bound and a simple and fast 2-approximation algorithm based on any interval model of the input graph.

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A Minimum Linear Arrangement Algorithm for Undirected Trees

Cited by 88 publications

References 3 publications

Approximation Algorithms for Minimizing Average Distortion

Approximation Algorithms for Minimizing Average Distortion

On Minimum Circular Arrangement

Optimal Linear Arrangement of Interval Graphs

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