## Fixed linear crossing minimization by reduction to the maximum cut problem

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Venue: | in Proc 12th Ann. Int. Computing and Combinatorics Conference (COCOON’06 |

Citations: | 6 - 0 self |

### BibTeX

@INPROCEEDINGS{Buchheim_fixedlinear,

author = {Christoph Buchheim and Lanbo Zheng},

title = {Fixed linear crossing minimization by reduction to the maximum cut problem},

booktitle = {in Proc 12th Ann. Int. Computing and Combinatorics Conference (COCOON’06},

year = {}

}

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### Abstract

Abstract. Many real-life scheduling, routing and location problems can be formulated as combinatorial optimization problems whose goal is to find a linear layout of an input graph in such a way that the number of edge crossings is minimized. In this paper, we study a restricted version of the linear layout problem where the order of vertices on the line is fixed, the so-called fixed linear crossing number problem (FLCNP). We show that this N P-hard problem can be reduced to the well-known maximum cut problem. The latter problem was intensively studied in the literature; efficient exact algorithms based on the branch-and-cut technique have been developed. By an experimental evaluation on a variety of graphs, we show that using this reduction for solving FLCNP compares favorably to earlier branch-and-bound algorithms. 1

### Citations

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(Show Context)
Citation Context ...or exact approaches instance B & B [1] MAXCUT Q4 0.01 0.00 CCC3 0.02 0.00 SX4 0.01 0.00 F LQ4 0.13 0.42 UDB5 0.43 0.07 C26(1, 3) 0.46 0.00 T6,6 1.27 0.04 CCC4 2.59 0.01 K10 2.27 3.21 SX5 2.16 1.84 C20=-=(1, 2, 3)-=- 16.69 0.39 T7,7 64.89 0.15 C22(1, 2, 3) 73.16 0.39 K11 148.21 24.56 Q5 612.35 1.67 K12 1925.51 79.15 K13 > 86400.00 2119.12 Notice that in our approach we did not use any initial heuristics, in order... |

214 |
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(Show Context)
Citation Context ...icircles above or below L; see Fig. 1. This type of drawing was first introduced by Nicholson [12] in order to develop a heuristic algorithm for the general N P-complete crossing minimization problem =-=[4]-=-. However, Masuda et al. proved that it is still N P-hard to find a linear embedding of a given graph with a minimum number of crossings, even if the ordering of vertices on L is predetermined [10]. T... |

143 | Succinct representation of balanced parentheses and static trees
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(Show Context)
Citation Context ... minimization for linear embeddings has important applications in different areas such as sorting permutations [6], fault tolerant VLSI design [13], complexity theory [3], and compact graph encodings =-=[11]-=-. Moreover, the problem FLCNP is of general interest in graph drawing and information visualization, where the number of edge crossings has a big effect on the readability of graph layout [2]. It was ... |

50 | Semidefinite programming and integer programming
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- 2005
(Show Context)
Citation Context ...known. More detailed information on algorithms for MAXCUT using cutting plane techniques can be found in [7, 9]. Observe that MAXCUT can also be adressed by semidefinite programming methods; see e.g. =-=[8]-=-. These methods perform well on very dense instances, while being outperformed by ILP approaches on sparse or large graphs. For this reason, we chose the latter method for our experiments. 4 Experimen... |

18 |
Crossing minimization in linear embeddings of graphs
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(Show Context)
Citation Context ...lem [4]. However, Masuda et al. proved that it is still N P-hard to find a linear embedding of a given graph with a minimum number of crossings, even if the ordering of vertices on L is predetermined =-=[10]-=-. The latter problem is called the fixed linear crossing number problem (FLCNP). Crossing minimization for linear embeddings has important applications in different areas such as sorting permutations ... |

16 |
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(Show Context)
Citation Context ... number problem (FLCNP). Crossing minimization for linear embeddings has important applications in different areas such as sorting permutations [6], fault tolerant VLSI design [13], complexity theory =-=[3]-=-, and compact graph encodings [11]. Moreover, the problem FLCNP is of general interest in graph drawing and information visualization, where the number of edge crossings has a big effect on the readab... |

10 | Algorithms for the fixed linear crossing number problem
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- 2002
(Show Context)
Citation Context ...to find a layout with no crossings at all. Recently, heuristic methods, as well as exact algorithms, have been proposed to find optimal or near-optimal solutions of linear layout problems. Cimikowski =-=[1]-=- presented different powerful heuristics as well as an exact branch-and-bound algorithm for FLCNP. In the worst case, the latter enumerates all possible assignments of edges to the two sides of L (up ... |

10 | Computing exact ground states of hard ising spin glass problems by branch-and-cut
- Liers, Jünger, et al.
- 2004
(Show Context)
Citation Context ...e MAXCUT problem has been investigated intensively, so that many classes of cutting planes are known. More detailed information on algorithms for MAXCUT using cutting plane techniques can be found in =-=[7, 9]-=-. Observe that MAXCUT can also be adressed by semidefinite programming methods; see e.g. [8]. These methods perform well on very dense instances, while being outperformed by ILP approaches on sparse o... |

7 |
The Max-Cut Problem
- Laurent
- 1997
(Show Context)
Citation Context ...e MAXCUT problem has been investigated intensively, so that many classes of cutting planes are known. More detailed information on algorithms for MAXCUT using cutting plane techniques can be found in =-=[7, 9]-=-. Observe that MAXCUT can also be adressed by semidefinite programming methods; see e.g. [8]. These methods perform well on very dense instances, while being outperformed by ILP approaches on sparse o... |

4 |
Forbidden subsequences and permutations sortable on two parallel stacks
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- 2001
(Show Context)
Citation Context .... The latter problem is called the fixed linear crossing number problem (FLCNP). Crossing minimization for linear embeddings has important applications in different areas such as sorting permutations =-=[6]-=-, fault tolerant VLSI design [13], complexity theory [3], and compact graph encodings [11]. Moreover, the problem FLCNP is of general interest in graph drawing and information visualization, where the... |

2 |
CNMgraf – graphic presentation services for network management
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- 1985
(Show Context)
Citation Context ...ere the number of edge crossings has a big effect on the readability of graph layout [2]. It was also shown to be a subproblem in communications network management graphics facilities such as CNMgraf =-=[5]-=-. Sorting with parallel stacks ⋆ Partially supported by the Marie Curie RTN ADONET 504438 funded by the EUsv 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 Fig. 1. A linear embedding is similar to FLCNP where the layo... |

2 |
circa 1986
- ROSENBERG, DIOGENES
- 1986
(Show Context)
Citation Context ...he fixed linear crossing number problem (FLCNP). Crossing minimization for linear embeddings has important applications in different areas such as sorting permutations [6], fault tolerant VLSI design =-=[13]-=-, complexity theory [3], and compact graph encodings [11]. Moreover, the problem FLCNP is of general interest in graph drawing and information visualization, where the number of edge crossings has a b... |

1 |
Permutation procedure for minimizing the number of crossings in a network
- Nicholson
- 1968
(Show Context)
Citation Context ...cial type of embedding in which vertices of V are placed on a horizontal line L and edges are drawn as semicircles above or below L; see Fig. 1. This type of drawing was first introduced by Nicholson =-=[12]-=- in order to develop a heuristic algorithm for the general N P-complete crossing minimization problem [4]. However, Masuda et al. proved that it is still N P-hard to find a linear embedding of a given... |