## Two-Layer Planarization in Graph Drawing (1998)

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Venue: | PROC. 9TH INTERNATIONAL SYMP. ON ALGORITHMS AND COMPUTATION (ISAAC'98), VOLUME 1533 OF LECTURE NOTES IN COMPUT. SCI |

Citations: | 6 - 0 self |

### BibTeX

@INPROCEEDINGS{Mutzel98two-layerplanarization,

author = {Petra Mutzel and René Weiskircher},

title = {Two-Layer Planarization in Graph Drawing},

booktitle = {PROC. 9TH INTERNATIONAL SYMP. ON ALGORITHMS AND COMPUTATION (ISAAC'98), VOLUME 1533 OF LECTURE NOTES IN COMPUT. SCI},

year = {1998},

pages = {69--78},

publisher = {Springer}

}

### Years of Citing Articles

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### Abstract

We study the two-layer planarization problems that have applications in Automatic Graph Drawing. We are searching for a two-layer planar subgraph of maximum weight in a given two-layer 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

### Citations

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The ellipsoid method and its consequences in combinatorial optimization, Combinatorica 1
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Citation Context ...omial time. Theorem 6. For the maximal clique inequalities, the separation problem can be solved inpolynomial time by computing at most jEj shortest path problems. According to earlier results (e.g., =-=[8]-=-), we can optimize a linear objective function over a polytope in polynomial time if and only if we can solve the separation problem in polynomial time. Hence, Theorem 6 gives us a polynomial time alg... |

210 |
Crossing number is NP–complete
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Citation Context ...ther one, such that the number of crossings is reduced. We suggest an alternative approach for the second step. Already for two-layer graphs the straight-line crossing minimization problem is NP-hard =-=[6]-=- even if one layer is xed [5]. Exact algorithms based on branchand bound have been suggested by various authors (see, e.g., [9]). For k 2, a vastsamount of heuristics has been published in the literat... |

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A cutting plane algorithm for the linear ordering problem
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Citation Context ...if vertex u is before v and 0 otherwise. The polytope Q1 = Q1(A� B� E� n A� ) = conv B j o B is a permutation of the vertices inB is identical to the linear ordering polytope that has been stu=-=died in [7]. -=-If we denote the points in P1 by (y� x), where y 2 R L , x 2 R jEj , we have the following relationship between the two polytopes P1 and Q1: Q1 = P1 \fx =0g. osThis fact will lead us to investigate ... |

85 |
Edge crossing in drawings of bipartite graphs
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- 1994
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Citation Context ...r of crossings is reduced. We suggest an alternative approach for the second step. Already for two-layer graphs the straight-line crossing minimization problem is NP-hard [6] even if one layer is xed =-=[5]-=-. Exact algorithms based on branchand bound have been suggested by various authors (see, e.g., [9]). For k 2, a vastsamount of heuristics has been published in the literature (see, e.g., [14] and [3])... |

46 |
Automatic display of hierarchized graphs for computer aided decision analysis
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Citation Context ...ion of structural information allows the reader to focus on the information content of the diagram. A common method for drawing directed graphs has been introduced by Sugiyama et al. [14] and Carpano =-=[2]-=-. In the rst step, the vertices are partitioned into a set of k layers, and in the second step, the vertices within each layer are permuted in such away that the number of crossings is small. In pract... |

39 |
Properties of vertex packing and independence system polyhedra
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Citation Context ...ct to set inclusion. An independence set is called k-regular if each of its circuits is of size k. The set F E is a clique of (E�I), ifjF j k and all ; jF j k-subsets k of F are circuits of (E�I).=-= In [12] iti-=-sshown that a maximal clique F E in a k-regular independence system (E�I) gives a facet-de ning inequality, namely, the clique inequality X xe k ; 1� (1) e2F for PI, the polytope associated with (... |

34 |
Heuristics for reducing crossings in 2-layered networks
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Citation Context ... [5]. Exact algorithms based on branchand bound have been suggested by various authors (see, e.g., [9]). For k 2, a vastsamount of heuristics has been published in the literature (see, e.g., [14] and =-=[3]-=-). A new approach isto remove a minimal set of edges such that the remaining k-layer graph can be drawn without edge crossings. In the nal drawing, the removed edges are reinserted. Since the insertio... |

32 |
Drawing graphs in two layers
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Citation Context ...(A� B� F� � ), F E, ofmaximum weight. To our knowledge, only the unweighted (we = 1 for all e 2 E) two-layer planarization problems have been considered in the literature so far. Eades and Whi=-=tesides [4]-=- showed NP-hardness for the latter two versions of the planarization problem and showed that the two layer xed version can be solved by transforming it to a longest increasing subsequence problem. The... |

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- 1997
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Citation Context ... subsequence problem, it can be solved in time O(jEj log jEj). Both theorems are not surprising, since there are similar results for the trace polytope T2 on two sequences that has been introduced in =-=[13] in the context of multi-=-ple sequence alignment. The set of circuits in the independence system IT (G) is S[ff(p� u)� (p� v)g j B(u) < B(v)� (p� u) 2 E�(p� v) 2 Eg [ff(p� u)� (q� u)g j A(p) < A(q)� (p�... |

17 | Exact and heuristic algorithms for 2layer straightline crossing minimization
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- 1995
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Citation Context ...-layer graphs the straight-line crossing minimization problem is NP-hard [6] even if one layer is xed [5]. Exact algorithms based on branchand bound have been suggested by various authors (see, e.g., =-=[9]-=-). For k 2, a vastsamount of heuristics has been published in the literature (see, e.g., [14] and [3]). A new approach isto remove a minimal set of edges such that the remaining k-layer graph can be d... |

14 |
On planarization algorithms of 2-level graphs
- Tomii, Kambayashi, et al.
- 1977
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Citation Context ...ns of the planarization problem and showed that the two layer xed version can be solved by transforming it to a longest increasing subsequence problem. The none layer xed version was rst mentioned in =-=[15]-=-. The authors introduced the problem in the context of graph drawing. Recently, the weighted two-layer planarization problem has been attacked, in which the layers are allowed to be permuted freely [1... |

6 |
The design of the branch and cut system ABACUS
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- 1997
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Citation Context ...e inequalities (2)-(4),(6)-(8), and (10) can be solved inpolynomial time. For the two-layer planarization problem with one xed layer, we implemented a branch-and-cut algorithm using the ABACUS-System =-=[10]-=-. Because of space limits, we cannot describe our branch-and-cut algorithm in more detail. We use separation routines for the inequalities given in Lemma 4 in order to get good upper bounds. Moreover,... |

3 |
On planarization algorithms of 2-level graphs
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- 1981
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Citation Context ...A good visualization of structural information allows the reader to focus on the information content of the diagram. A common method for drawing directed graphs has been introduced by Sugiyama et al. =-=[14]-=- and Carpano [2]. In the rst step, the vertices are partitioned into a set of k layers, and in the second step, the vertices within each layer are permuted in such away that the number of crossings is... |

2 |
An alternative approach for drawing hierarchical graphs
- Mutzel
- 1997
(Show Context)
Citation Context ...aph. Depending on the number of layers in which the vertices can be permuted freely, that is, zero, one or two, di erent versions of the problems arise. The latter problem was already investigated in =-=[11]-=- using polyhedral combinatorics. Here, we study the remaining two cases and the relationships between the associated polytopes. In particular, we investigate the polytope P1 associated with the twolay... |

1 |
Drawing directed acyclic graphs: An experimental study (preliminary version
- Battista, Garg, et al.
- 1996
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Citation Context ...problem when both layers are xed. 6 Computational Results To test the performance of our branch-and-cut algorithm for the two-layer planarization problem (1 layer xed), we worked with the graphs from =-=[1]-=- that are called the North DAGs. These directed acyclic graphs have 10 to 100 nodes. We distributed them into sets Gi with i running form 1 to 9 such that the set Gi holds the graphs where the number ... |