A New Approach to Exact Crossing Minimization

Chimani, Markus; Mutzel, Petra; Bomze, Immanuel M.

doi:10.1007/978-3-540-87744-8_24

Cited by 34 publications

(27 citation statements)

References 16 publications

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“…On the other hand, it has been shown that the problem is fixed parameter tractable: One can test whether a graph has a crossing number at most k in linear time, when considering k fixed [14,23]. While these approaches do currently not allow practical algorithms, there exist linear programming based exact algorithms that are promising for "real-world" graphs arising in graph drawing applications [8]. Yet, computing exact crossing numbers is in general extremely difficult and one usually has to resort to heuristics, see, e.g., [1,15].…”

Section: Computational Complexitymentioning

confidence: 99%

Approximating the Crossing Number of Graphs Embeddable in Any Orientable Surface

Hliněný¹,

Chimani²

2010

Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms

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The crossing number of a graph is the least number of pairwise edge crossings in a drawing of the graph in the plane. We provide an O(n log n) time constant factor approximation algorithm for the crossing number of a graph of bounded maximum degree which is "densely enough" embeddable in an arbitrary fixed orientable surface.Our approach combines some known tools with a powerful new lower bound on the crossing number of an embedded graph. This result extends previous results that gave such approximations in particular cases of projective, toroidal or apex graphs; it is a qualitative improvement over previously published algorithms that constructed low-crossing-number drawings of embeddable graphs without giving any approximation guarantees. No constant factor approximation algorithms for the crossing number problem over comparably rich classes of graphs are known to date.

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Section: Computational Complexitymentioning

confidence: 99%

Approximating the Crossing Number of Graphs Embeddable in Any Orientable Surface

Hliněný¹,

Chimani²

2010

Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms

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“…Therefore, to estimate the limit within given precision ε, it suffices to determine the constant N of Theorem and compute the crossing number of

⊙ (T^{N})

, which can be done using the algorithm of Chimani et al. . The resulting algorithm is exponential in a polynomial of

\frac{1}{ε}

.…”

Section: Introductionmentioning

confidence: 99%

Crossing Numbers of Periodic Graphs

Dvořák¹,

Mohar

2015

Journal of Graph Theory

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“…(iii) One can study an ILP formulation for the problem. To this aim, the crossing minimization strategy in [20] may provide useful insights to compute crossing minimal 1-planar (or more in general k-planar) embeddings.…”

Section: Conclusion and Future Research Directionsmentioning

confidence: 99%

An Experimental Study of a 1-Planarity Testing and Embedding Algorithm

Binucci

Didimo

Montecchiani

2020

WALCOM: Algorithms and Computation

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The definition of 1-planar graphs naturally extends graph planarity, namely a graph is 1-planar if it can be drawn in the plane with at most one crossing per edge. Unfortunately, while testing graph planarity is solvable in linear time, deciding whether a graph is 1-planar is NP-complete, even for restricted classes of graphs. Although several polynomial-time algorithms have been described for recognizing specific subfamilies of 1-planar graphs, no implementations of general algorithms are available to date. We investigate the feasibility of a 1-planarity testing and embedding algorithm based on a backtracking strategy. While the experiments show that our approach can be successfully applied to graphs with up to 30 vertices, they also suggest the need of more sophisticated techniques to attack larger graphs. Our contribution provides initial indications that may stimulate further research on the design of practical approaches for the 1-planarity testing problem.

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A New Approach to Exact Crossing Minimization

Cited by 34 publications

References 16 publications

Approximating the Crossing Number of Graphs Embeddable in Any Orientable Surface

Approximating the Crossing Number of Graphs Embeddable in Any Orientable Surface

Crossing Numbers of Periodic Graphs

An Experimental Study of a 1-Planarity Testing and Embedding Algorithm

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