## 2-Layer Straightline Crossing Minimization: Performance of Exact and Heuristic Algorithms (1997)

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Citations: | 69 - 6 self |

### BibTeX

@MISC{Jünger972-layerstraightline,

author = {Michael Jünger and Petra Mutzel},

title = {2-Layer Straightline Crossing Minimization: Performance of Exact and Heuristic Algorithms},

year = {1997}

}

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### Abstract

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 NP-hard, 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.

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Citation Context ...e information content of the diagram. Examples are entity-relationship diagrams, PERT-diagrams, or any flow diagram. A common method for drawing directed graphs has been introduced by Sugiyama et al. =-=[STT81]-=- and Carpano [Car80]. In the first step, the vertices are partitioned into a set of k levels, and in the second step, the vertices within each level are permuted in such a way that the number of cross... |

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Citation Context ... are drawn as straight lines. Essentially, a k-layered network is a k-partite graph that is drawn in a special way. Even for 2-layered graphs the straightline crossing minimization problem is NP-hard =-=[GJ83]-=-. The problem consists of aligning the two shores V 1 and V 2 of a bipartite graph G = (V 1 ; V 2 ; E) on two parallel straight lines (layers) such that the number of crossings between the edges in E ... |

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Citation Context ...entage of the minimum number of crossings, and the average running time in seconds on a SUN Sparcstation 10. All samples are generated by the program random bigraph of the Stanford GraphBase by Knuth =-=[K93]. The-=- generators are hardware independent and are available from the authors so that exactly the same experiments can be run by anyone who is interested. In Figure 2, we give the results for "20+20-gr... |

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Citation Context ... only 2 \Gamma n 2 \Delta hypercube inequalities and 2 \Gamma n 3 \Delta 3-cycle inequalities are left that define a relaxation of (LO) which has been proven very useful in practical applications. In =-=[GJR84a]-=- a branch and cut algorithm for (LO) is proposed that solves this relaxation with a cutting plane approach, since writing down all 3-cycle inequalities, even though taking only polynomial space, and s... |

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Citation Context ...of one layer xed is also NP-hard [7]. Therefore, a lot of e ort went into the design of e cient heuristics, for the version in which one permutation is xed as well as for the general case (see, e.g., =-=[16, 14,6,8,4,2]-=- and [15]). Eades and Kelly [6] observe that the computation of true optima would be desirable in order to assess the performance of various heuristics, however, they believe that the NP-hardness of t... |

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Citation Context ...t to increasing time is barycenter, median, greedy-insert, assign, greedy-switch, stochastic, exact, and split. In Table 1, we repeat an experiment by Dresbach [D94] for instances defined by Warfield =-=[W77]-=- as follows: For k = 3; 4; 5; 6; 7; 8 we let n 1 = k, n 2 = 2 k \Gamma 1, and the adjacency matrix of the 5 99.5 100 100.5 101 101.5 102 102.5 103 103.5 104 104.5 105 105.5 106 106.5 107 107.5 108 108... |

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Citation Context ...content of the diagram. Examples are entity-relationship diagrams, PERTdiagrams, or any ow diagram. A common method for drawing directed graphs has been introduced by Sugiyama et al. [14] and Carpano =-=[1]-=-. In the rst step, the vertices are partitioned into a set of k levels, and in the second step, the vertices within each level are permuted in such away that the number of crossings is small. In this ... |

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Citation Context ...went into the design of efficient heuristics, for the version in which one permutation is fixed as well as for the general case (see, e.g., [W77,STT81,EK86,EW94,D94,C95] and [VML96]). Eades and Kelly =-=[EK86]-=- observe that the computation of true optima would be desirable in order to assess the performance of various heuristics, however, [EK86] believe that the NP-hardness of the problem renders such an ex... |

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Citation Context ....sM. Junger et al., 2-Layer Crossing Minimization, JGAA, 1(1) 1{25 (1997) 3 Unfortunately, the 2-layer straightline crossing minimization problem with the permutation of one layer xed is also NP-hard =-=[7]-=-. Therefore, a lot of e ort went into the design of e cient heuristics, for the version in which one permutation is xed as well as for the general case (see, e.g., [16, 14,6,8,4,2] and [15]). Eades an... |

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Citation Context ... with a number of members exponential in n that must be present in a complete linear description of the feasible set, are known, and some of them can be exploited algorithmically. For the details see =-=[GJR85]-=-. When the integrality conditions in (LO) are dropped, only 2 \Gamma n 2 \Delta hypercube inequalities and 2 \Gamma n 3 \Delta 3-cycle inequalities are left that define a relaxation of (LO) which has ... |

15 | The assignment heuristic for crossing reduction
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Citation Context ...median heuristic by [EW94], the stochastic heuristic by [D94], the greedy-insert heuristic by [EK86], the greedy-switch heuristic by [EK86], the split heuristic by [EK86], and the assign heuristic by =-=[C95]. The barycenter heu-=-ristic [STT81] and the median heuristic [EW94] are the most popular ones. They are also called "averaging heuristics", since they simply compute the "average position", i.e., the b... |

12 |
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Citation Context ...herefore, a lot of effort went into the design of efficient heuristics, for the version in which one permutation is fixed as well as for the general case (see, e.g., [W77,STT81,EK86,EW94,D94,C95] and =-=[VML96]-=-). Eades and Kelly [EK86] observe that the computation of true optima would be desirable in order to assess the performance of various heuristics, however, [EK86] believe that the NP-hardness of the p... |

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Citation Context ...d the inclusion of further useful inequalities in the cutting plane part, see [GJR84a]. A new implementation of the algorithm is used in our computational experiments. It is written in C and uses the =-=[CPLEX]-=- software for solving the linear programming relaxations coming up in the course of the computation. 3. One Sided Crossing Minimization The fact that we are able to compute optimum solutions allows us... |

10 |
A New Heuristic Layout Algorithm for DAGS
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Citation Context ... crossing minimization experimentally. Our computational comparison includes the following heuristics: the barycenter heuristic by [STT81], the median heuristic by [EW94], the stochastic heuristic by =-=[D94]-=-, the greedy-insert heuristic by [EK86], the greedy-switch heuristic by [EK86], the split heuristic by [EK86], and the assign heuristic by [C95]. The barycenter heuristic [STT81] and the median heuris... |

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Citation Context ...the inclusion of further useful inequalities in the cutting plane part, see [10]. A new implementation of the algorithm is used in our computational experiments. It is written in C and uses the CPLEX =-=[3]-=- software for solving the linear programming relaxations coming up in the course of the computation. 3 One Sided Crossing Minimization The fact that we are able to compute optimum solutions allows us ... |

7 |
Optimal triangulation of large real world input-output matrices. Statistische Hefte
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Citation Context ...is solved. If the optimum solution is integral, the algorithm stops, otherwise it is applied recursively to two subproblems in one of which a fractional x ij is set to 1 and in the other set to 0. In =-=[GJR84b]-=- such a branch and cut approach could be used to find optimum linear orderings with n up to 60 in an application involving input-output matrices that are used in economic analysis. For the many detail... |

6 |
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Citation Context ...of one layer xed is also NP-hard [7]. Therefore, a lot of e ort went into the design of e cient heuristics, for the version in which one permutation is xed as well as for the general case (see, e.g., =-=[16, 14,6,8,4,2]-=- and [15]). Eades and Kelly [6] observe that the computation of true optima would be desirable in order to assess the performance of various heuristics, however, they believe that the NP-hardness of t... |

2 |
Edge Crossings in Drawings of Bipartite Graphs", manuscripts
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Citation Context ...sists of repeating these two loops until no more improvement is obtained. Unfortunately, the 2-layer straightline crossing minimization problem with the permutation of one layer fixed is also NP-hard =-=[EW94]-=-. Therefore, a lot of effort went into the design of efficient heuristics, for the version in which one permutation is fixed as well as for the general case (see, e.g., [W77,STT81,EK86,EW94,D94,C95] a... |

1 |
Personal communication
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Citation Context ...ely. Also the results in [C95] on the assign 4 heuristic are in line with ours. There are no published computational results for the stochastic heuristic, but a personal communication with the author =-=[D95]-=- confirms the correctness of our implementation. All subsequent figures and tables use the following notation: -- n i : Number of nodes on layer i for i = 1; 2 -- m: Number of edges -- Lowerbound: The... |