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30
2Layer Straightline Crossing Minimization: Performance of Exact and Heuristic Algorithms
, 1997
"... We present algorithms for the two layer straightline crossing minimization problem that are able to compute exact optima. Our computational results lead us to the conclusion that there is no need for heuristics if one layer is fixed, even though the problem is NPhard, and that for the general probl ..."
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Cited by 64 (6 self)
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We present algorithms for the two layer straightline crossing minimization problem that are able to compute exact optima. Our computational results lead us to the conclusion that there is no need for heuristics if one layer is fixed, even though the problem is NPhard, and that for the general problem with two variable layers, true optima can be computed for sparse instances in which the smaller layer contains up to 15 nodes. For bigger instances, the iterated barycenter method turns out to be the method of choice among several popular heuristics whose performance we could assess by comparing their results to optimum solutions.
On Linear Layouts of Graphs
, 2004
"... In a total order of the vertices of a graph, two edges with no endpoint in common can be crossing, nested, or disjoint. A kstack (resp... ..."
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Cited by 30 (18 self)
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In a total order of the vertices of a graph, two edges with no endpoint in common can be crossing, nested, or disjoint. A kstack (resp...
On the Parameterized Complexity of Layered Graph Drawing
 PROC. 5TH ANNUAL EUROPEAN SYMP. ON ALGORITHMS (ESA '01
, 2001
"... We consider graph drawings in which vertices are assigned to layers and edges are drawn as straight linesegments between vertices on adjacent layers. We prove that graphs admitting crossingfree hlayer drawings (for fixed h) have bounded pathwidth. We then use a path decomposition as the basis for ..."
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Cited by 20 (8 self)
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We consider graph drawings in which vertices are assigned to layers and edges are drawn as straight linesegments between vertices on adjacent layers. We prove that graphs admitting crossingfree hlayer drawings (for fixed h) have bounded pathwidth. We then use a path decomposition as the basis for a lineartime algorithm to decide if a graph has a crossingfree hlayer drawing (for fixed h). This algorithm is extended to solve a large number of related problems, including allowing at most k crossings, or removing at most r edges to leave a crossingfree drawing (for fixed k or r). If the number of crossings or deleted edges is a nonfixed parameter then these problems are NPcomplete. For each setting, we can also permit downward drawings of directed graphs and drawings in which edges may span multiple layers, in which case the total span or the maximum span of edges can be minimized. In contrast to the socalled Sugiyama method for layered graph drawing, our algorithms do not assume a preassignment of the vertices to layers.
Heuristics and Experimental Design for Bigraph Crossing Number Minimization
 IN ALGORITHM ENGINEERING AND EXPERIMENTATION (ALENEX’99), NUMBER 1619 IN LECTURE NOTES IN COMPUTER SCIENCE
, 1999
"... The bigraph crossing problem, embedding the two vertex sets of a bipartite graph G = (V0;V1;E) along two parallel lines so that edge crossings are minimized, has application to circuit layout and graph drawing. We consider the case where both V0 and V1 can be permuted arbitrarily  both this and ..."
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Cited by 14 (9 self)
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The bigraph crossing problem, embedding the two vertex sets of a bipartite graph G = (V0;V1;E) along two parallel lines so that edge crossings are minimized, has application to circuit layout and graph drawing. We consider the case where both V0 and V1 can be permuted arbitrarily  both this and the case where the order of one vertex set is fixed are NPhard. Two new heuristics that perform well on sparse graphs such as occur in circuit layout problems are presented. The new heuristics outperform existing heuristics on graph classes that range from applicationspecific to random. Our experimental design methodology ensures that differences in performance are statistically significant and not the result of minor variations in graph structure or input order.
Local search with very largescale neighborhoods for optimal permutations in machine translation
 In Proc. of the Workshop on Computationally Hard Problems and Joint Inference
, 2006
"... We introduce a novel decoding procedure for statistical machine translation and other ordering tasks based on a family of Very LargeScale Neighborhoods, some of which have previously been applied to other NPhard permutation problems. We significantly generalize these problems by simultaneously con ..."
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Cited by 12 (3 self)
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We introduce a novel decoding procedure for statistical machine translation and other ordering tasks based on a family of Very LargeScale Neighborhoods, some of which have previously been applied to other NPhard permutation problems. We significantly generalize these problems by simultaneously considering three distinct sets of ordering costs. We discuss how these costs might apply to MT, and some possibilities for training them. We show how to search and sample from exponentially large neighborhoods using efficient dynamic programming algorithms that resemble statistical parsing. We also incorporate techniques from statistical parsing to improve the runtime of our search. Finally, we report results of preliminary experiments indicating that the approach holds promise. 1
Comparing trees via crossing minimization
 In Proc. 25th Conf. on Foundations of Software Technology and Theoretical Computer Science, volume 3821 of LNCS
, 2005
"... Abstract. Two trees with the same number of leaves have to be embedded in two layers in the plane such that the leaves are aligned in two adjacent layers. Additional matching edges between the leaves give a onetoone correspondence between pairs of leaves of the different trees. Do there exist two ..."
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Cited by 11 (1 self)
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Abstract. Two trees with the same number of leaves have to be embedded in two layers in the plane such that the leaves are aligned in two adjacent layers. Additional matching edges between the leaves give a onetoone correspondence between pairs of leaves of the different trees. Do there exist two planar embeddings of the two trees that minimize the crossings of the matching edges? This problem has important applications in the construction and evaluation of phylogenetic trees.
An Efficient Fixed Parameter Tractable Algorithm for 1Sided Crossing Minimization
 ALGORITHMICA
, 2004
"... We give an O(ϕ k · n 2) fixed parameter tractable algorithm for the 1SIDED CROSSING MINIMIZATION problem. The constant ϕ in the running time is the golden ratio ϕ = (1 + √ 5)/2 ≈ 1.618. The constant k is the parameter of the problem: the number of allowed edge crossings. ..."
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Cited by 11 (4 self)
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We give an O(ϕ k · n 2) fixed parameter tractable algorithm for the 1SIDED CROSSING MINIMIZATION problem. The constant ϕ in the running time is the golden ratio ϕ = (1 + √ 5)/2 ≈ 1.618. The constant k is the parameter of the problem: the number of allowed edge crossings.
A FixedParameter Approach to TwoLayer Planarization
, 2002
"... A bipartite graph is biplanar if the vertices can be placed on two parallel lines (layers) in the plane such that there are no edge crossings when edges are drawn straight. The 2Layer Planarization problem asks if k edges can be deleted from a given graph G so that the remaining graph is biplanar ..."
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Cited by 11 (3 self)
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A bipartite graph is biplanar if the vertices can be placed on two parallel lines (layers) in the plane such that there are no edge crossings when edges are drawn straight. The 2Layer Planarization problem asks if k edges can be deleted from a given graph G so that the remaining graph is biplanar. This problem is NPcomplete, as is the 1Layer Planarization problem in which the permutation of the vertices in one layer is fixed. We give the following fixed parameter tractability results: an O(k ·6 k +G) algorithm for 2Layer Planarization and an O(3 k ·G) algorithm for 1Layer Planarization, thus achieving linear time for fixed k.
DAG Drawing from an Information Visualization Perspective
, 1999
"... When dealing with a graph, any visualization strategy must rely on a layout procedure at least to initiate the process. Because the visualization process evolves within an interactive environment the choice of this layout procedure is critical and will often be based on efficiency. This paper com ..."
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Cited by 7 (2 self)
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When dealing with a graph, any visualization strategy must rely on a layout procedure at least to initiate the process. Because the visualization process evolves within an interactive environment the choice of this layout procedure is critical and will often be based on efficiency. This paper compares two popular layout strategies, one based on the extraction of a spanning tree, the other based on edge crossing minimization of directed acyclic graphs. The comparison is made based on a large number of experimental evidence gathered through random graph generation. The main conclusion of these experiments is that, contrary to the popular belief, usage of edge crossing minimization algorithms may be extremely useful and advantageous, even under the heavy requirements of information visualization.
TwoLayer Planarization in Graph Drawing
 PROC. 9TH INTERNATIONAL SYMP. ON ALGORITHMS AND COMPUTATION (ISAAC'98), VOLUME 1533 OF LECTURE NOTES IN COMPUT. SCI
, 1998
"... We study the twolayer planarization problems that have applications in Automatic Graph Drawing. We are searching for a twolayer planar subgraph of maximum weight in a given twolayer graph. Depending on the number of layers in which the vertices can be permuted freely, that is, zero, one or tw ..."
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Cited by 6 (0 self)
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We study the twolayer planarization problems that have applications in Automatic Graph Drawing. We are searching for a twolayer planar subgraph of maximum weight in a given twolayer graph. Depending on the number of layers in which the vertices can be permuted freely, that is, zero, one or two, different versions of the problems arise. The latter problem was already investigated in [11] using polyhedral combinatorics. Here, we study