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A FixedParameter Approach to 2Layer Planarization
, 2006
"... A bipartite graph is biplanar if the vertices can be placed on two parallel lines (layers) inthe plane such that there are no edge crossings when edges are drawn as line segments between the layers. In this paper we study the 2LAYER PLANARIZATION problem: Can k edges be deleted from a given graph ..."
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Cited by 9 (2 self)
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A bipartite graph is biplanar if the vertices can be placed on two parallel lines (layers) inthe plane such that there are no edge crossings when edges are drawn as line segments between the layers. In this paper we study the 2LAYER PLANARIZATION problem: Can k edges be deleted from a given graph G so that the remaining graph is biplanar? This problem is NPcomplete, and remains so if the permutation of the vertices in one layer is fixed (the 1LAYER PLANARIZATION problem). We prove that these problems are fixedparameter tractable by giving lineartime algorithms for their solution (for fixed k). In particular, we solve the 2LAYER PLANARIZATION problem in O(k · 6 k +G) time and the 1LAYER PLANARIZATION problem in O(3 k ·G) time. We also show that there are polynomialtime constantapproximation algorithms for both problems.
A fast and simple heuristic for constrained twolevel crossing reduction
 In GD ’04
, 2004
"... Abstract. The onesided twolevel crossing reduction problem is an important problem in hierarchical graph drawing. Because of its NPhardness there are many heuristics, such as the wellknown barycenter and median heuristics. We consider the constrained onesided twolevel crossing reduction proble ..."
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Cited by 7 (0 self)
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Abstract. The onesided twolevel crossing reduction problem is an important problem in hierarchical graph drawing. Because of its NPhardness there are many heuristics, such as the wellknown barycenter and median heuristics. We consider the constrained onesided twolevel crossing reduction problem, where the relative position of certain vertex pairs on the second level is fixed. Based on the barycenter heuristic, we present a new algorithm that runs in quadratic time and generates fewer crossings than existing simple extensions. It is significantly faster than an advanced algorithm by Schreiber [12] and Finnocchi [1, 2, 6], while it compares well in terms of crossing number. It is also easy to implement. 1
On the Complexity of Crossings in Permutations
, 2007
"... We investigate crossing minimization problems for a set of permutations, where a crossing expresses a disarrangement between elements. The goal is a common permutation π ∗ which minimizes the number of crossings. In voting and social science theory this is known as the Kemeny optimal aggregation pro ..."
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Cited by 6 (1 self)
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We investigate crossing minimization problems for a set of permutations, where a crossing expresses a disarrangement between elements. The goal is a common permutation π ∗ which minimizes the number of crossings. In voting and social science theory this is known as the Kemeny optimal aggregation problem minimizing the Kendallτ distance. This rank aggregation problem can be phrased as a onesided twolayer crossing minimization problem for a series of bipartite graphs or for an edge coloured bipartite graph, where crossings are counted only for monochromatic edges. We contribute the max version of the crossing minimization problem, which attempts to minimize the discrimination against any permutation. As our results, we correct the construction from [9] and prove the NPhardness of the common crossing minimization problem for k = 4 permutations. Then we establish a 2 − 2/kapproximation, improving the previous factor of 2. The max version is shown NPhard for every k ≥ 4, and there is a 2approximation. Both approximations are
The Barycenter Heuristic and the Reorderable Matrix
, 2005
"... this paper is to discuss the role of the barycenter heuristic in ordering the rows and columns of the matrix. So far, the barycenter heuristic has been mainly used in graph drawing algorithms. In order to gain full advantage of the barycenter heuristic in ordering rows and columns of the reorderable ..."
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Cited by 4 (0 self)
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this paper is to discuss the role of the barycenter heuristic in ordering the rows and columns of the matrix. So far, the barycenter heuristic has been mainly used in graph drawing algorithms. In order to gain full advantage of the barycenter heuristic in ordering rows and columns of the reorderable matrix, we survey its use in various contexts and recall the theoretical results obtained
Optimal Partition of a Bipartite Graph into Non–Crossing b–Matchings”, manuscript
, 2001
"... Given a bipartite graph and a layout of it, we address the problem of partitioning the edge set of the graph into the minimum number of non–crossing matchings, that is subsets of edges no two of which share a common vertex or cross each other in the plane. We discuss some lower and upper bounds on t ..."
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Cited by 3 (0 self)
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Given a bipartite graph and a layout of it, we address the problem of partitioning the edge set of the graph into the minimum number of non–crossing matchings, that is subsets of edges no two of which share a common vertex or cross each other in the plane. We discuss some lower and upper bounds on the minimum number of classes of such a partition into non–crossing matchings, and devise an exact almost linear algorithm.
How to Rank with Fewer Errors  A PTAS for Feedback Arc Set in Tournaments
"... We present the first polynomial time approximation scheme (PTAS) for the minimum feedback arc set problem in tournaments. A weighted generalization gives the first PTAS for Kemeny rank aggregation. The runtime is singly exponential in 1/ǫ, improving on the conference version of this work, which was ..."
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We present the first polynomial time approximation scheme (PTAS) for the minimum feedback arc set problem in tournaments. A weighted generalization gives the first PTAS for Kemeny rank aggregation. The runtime is singly exponential in 1/ǫ, improving on the conference version of this work, which was doubly exponential.
TieBreaking Heuristics for the Barycenter and Median Algorithms
, 2006
"... The barycenter and median algorithms are well known practical methods to minimize the number of edge crossing in the twosided graph drawing problem for bipartite graphs. In this paper we study tiebreaking heuristics for these algorithms. Our experiments show that the tiebreaking heuristics inde ..."
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The barycenter and median algorithms are well known practical methods to minimize the number of edge crossing in the twosided graph drawing problem for bipartite graphs. In this paper we study tiebreaking heuristics for these algorithms. Our experiments show that the tiebreaking heuristics indeed improve the standard algorithms and that there are clear differences in the solution quality depending on how we break ties appearing in the algorithms. Our experiments also reveal the great amount of ties appearing especially in the case of the median algorithm. This explains the wellknown superiority of the barycenter algorithm over the median algorithm.