## Approximating Layout Problems on Random Geometric Graphs (2001)

Venue: | Journal of Algorithms |

Citations: | 21 - 10 self |

### BibTeX

@ARTICLE{Díaz01approximatinglayout,

author = {Josep Díaz and Mathew D. Penrose and Jordi Petit and María Serna},

title = {Approximating Layout Problems on Random Geometric Graphs},

journal = {Journal of Algorithms},

year = {2001},

volume = {39},

pages = {2001}

}

### Years of Citing Articles

### OpenURL

### Abstract

In this paper, we study the approximability of several layout problems on a family of random geometric graphs. Vertices of random geometric graphs are randomly distributed on the unit square and are connected by edges whenever they are closer than some given parameter. The layout problems that we consider are: Bandwidth, Minimum Linear Arrangement, Minimum Cut Width, Minimum Sum Cut, Vertex Separation and Edge Bisection. We first prove that some of these problems remain NP-complete even for geometric graphs. Afterwards, we compute lower bounds that hold, almost surely, for random geometric graphs. Then, we present two heuristics that, almost surely, turn to be constant approximation algorithms for our layout problems on random geometric graphs. In fact, for the Bandwidth and Vertex Separation problems, these heuristics are asymptotically optimal. Finally, we use the theoretical results in order to empirically compare these and other well-known heuristics. # This research was partially ...

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Citation Context ...p represents the probability of the existence of each possible edge. Random graphs have received much attention and together with the probabilistic method have become a powerful tool in combinatorics =-=[3, 10, 38]-=-. Partitioning properties for Gn,p graphs are studied in [11]. Layout problems for sparse random graphs are studied in [24], where it is shown that for each one of the considered layout measures, with... |

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Citation Context ... between the maximum and minimum values is a constant. 2sTherefore, the Gn,p model does not provide an informative framework to compare heuristics for layout problems. Geometric graphs or disk graphs =-=[13, 18]-=- have been proposed as a possible model to take into account the structural characteristics of instances that appear in most of the practical applications. Disk graphs are intersection graphs of disks... |

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Citation Context ...p represents the probability of the existence of each possible edge. Random graphs have received much attention and together with the probabilistic method have become a powerful tool in combinatorics =-=[3, 10, 38]-=-. Partitioning properties for Gn,p graphs are studied in [11]. Layout problems for sparse random graphs are studied in [24], where it is shown that for each one of the considered layout measures, with... |

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Citation Context ...case of the Bandwidth and Vertex Separation problems, the solutions returned by either of our algorithms are asymptotically optimal. In these cases, our result is an analog of the seminal BHH theorem =-=[6]-=- on the TSP tour of a random set of points in the unit square (Theorem 6.3). For the remaining problems, the approximation factor of the values provided by the two algorithms are tight. We emphasize t... |

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Citation Context ... clone fingerprinting [44]. Because of its importance in the design of divide-and-conquer algorithms for network problems and graph drawing, the Edge Bisection problem has received a lot of attention =-=[8, 51, 61]-=-. In the present paper we shall deal with the simpler non-weighted versions of those problems. An important characteristic of these applications is the specificity of their instances: for most applica... |

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Citation Context ...tion of edges in interconnection networks [16]. The Minimum Linear Arrangement problem, also called the Edgesum problem, is relevant in circuit and VLSI layout [72, 35], single machine job scheduling =-=[1, 70]-=- and as a simplified model for nervous system simulation [54]. The Minimum Cut Width problem was first used as a theoretical model for the number of channels in an optimal layout of a circuit [53, 2].... |

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Citation Context ...s between any pair of distinct nodes exist when their distance is smaller than some parameter r. Many empirical studies have used random models of geometric graphs for layout or partitioning problems =-=[39, 7, 49, 67]-=-. However, the theoretical study of random geometric graphs has been mainly focused on parameters as their clique or chromatic number and their connectivity properties (see [22] for a survey). Therefo... |

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Citation Context ... between the maximum and minimum values is a constant. 2sTherefore, the Gn,p model does not provide an informative framework to compare heuristics for layout problems. Geometric graphs or disk graphs =-=[13, 18]-=- have been proposed as a possible model to take into account the structural characteristics of instances that appear in most of the practical applications. Disk graphs are intersection graphs of disks... |

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Citation Context ...oice, the construction of sparse but connected graphs is guaranteed: Defining the connectivity distance ρn = ρn(Xn) as the smallest radius � 1 r such that the graph G(Xn, r) is connected, it is kn=-=own [4] that -=-ρn n/log n converges to 2 almost surely; that is, with probability 1 the sequence (Xi)i≥1 yields a sequence (ρn)n≥1 with this limit behavior. an = nr 2 n interacting. For results on other cases,... |

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Citation Context ...ost surely; that is, with probability 1 the sequence (Xi)i≥1 yields a sequence (ρn)n≥1 with this limit behavior. an = nr 2 n interacting. For results on other cases, for example with an constant,=-= see [20, 65]-=-. The parameter can be thought of as the local density of points on the scale at which they are 6s3 Complexity results for geometric instances In this section, we shall consider the decisional counter... |

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Citation Context ...ng problems [39, 7, 49, 67]. However, the theoretical study of random geometric graphs has been mainly focused on parameters as their clique or chromatic number and their connectivity properties (see =-=[22]-=- for a survey). Therefore, the analysis of layout measures in random geometric graphs certainly seems worthwhile to study. In this paper, we are concerned with the complexity and approximability of se... |

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Citation Context ...esting to understand how the optimal costs of our problems behave for different radii; see [20, 65]. Bounds on the cost of layout problems for lattice graphs and random lattice graphs can be found in =-=[21]-=-. Finally, we leave as an open problem to determine whether or not the Projection and Dissection algorithms are, in some probabilistic and asymptotic sense, optimal for layout problems other than Band... |

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Citation Context ...actical relevance in different applications. For instance, the Bandwidth problem is an important problem in matrix theory and is very much related to the dilation of edges in interconnection networks =-=[16]-=-. The Minimum Linear Arrangement problem, also called the Edgesum problem, is relevant in circuit and VLSI layout [72, 35], single machine job scheduling [1, 70] and as a simplified model for nervous ... |