Layer-free upward crossing minimization

Chimani, Markus; Gutwenger, Carsten; Mutzel, Petra; Wong, Hoi-Ming

doi:10.1145/1671970.1671975

Cited by 23 publications

(31 citation statements)

References 12 publications

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“…When drawing DAGs in practice, one usually tries to minimize crossings. For sparse graphs (thus including the Rome and North instance), the Layer Free Upward Planarization method (LFUP) [7] is known to be the currently strongest approach [1]. Using our new tool, we can ask how often LFUP achieves the optimum upward crossing number of 0 for upward planar graphs (Table 2).…”

Section: Methodsmentioning

confidence: 99%

“…In [3], an (exponential time) algorithm based on branch-and-bound over the possible embeddings was presented, which is able to decide upward planarity in reasonable time for sparse graphs with up to 200 vertices. This lack of practically applicable algorithms is especially unfortunate, as they would not only be interesting for their core functionality, but also to improve different building blocks of other algorithms in the realm of upward drawings: For example, the upward planarization approach [7] has to start with a large upward planar subgraph. Due to the lack of fast upward planarity testing algorithms, it uses relatively weak heuristics to find some upward planar subgraph which is of reasonable size.…”

Section: Introductionmentioning

confidence: 99%

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Upward Planarity Testing via SAT

Chimani

Zeranski

2013

Graph Drawing

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Abstract.A directed acyclic graph is upward planar if it allows a drawing without edge crossings where all edges are drawn as curves with monotonously increasing y-coordinates. The problem to decide whether a graph is upward planar or not is NP-complete in general, and while special graph classes are polynomial time solvable, there is not much known about solving the problem for general graphs in practice. The only attempt so far was a branch-and-bound algorithm over the graph's triconnectivity structure which was able to solve sparse graphs.In this paper, we propose a fundamentally different approach, based on the seemingly novel concept of ordered embeddings. We carefully model the problem as a special SAT instance, i.e., a logic formula for which we check satisfiability. Solving these SAT instances allows us to decide upward planarity for arbitrary graphs. We then show experimentally that this approach seems to dominate the known alternative approaches and is able to solve traditionally used graph drawing benchmarks effectively.

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Section: Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Upward Planarity Testing via SAT

Chimani

Zeranski

2013

Graph Drawing

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“…In contrast to this, the graphs leveled by the UPL approach only allow much smaller improvements. In fact, it shows that the upward planarization approach [3] gives near-optimal solutions for its respective leveling. We also observe that the fact that UPL produces more but smaller levels and requires less crossings is beneficial for both exact approaches: they solve all UPL instances, while the GKNV instances are harder.…”

Section: Real-world Graphsmentioning

confidence: 95%

“…UPL. Recent algorithms have combined the first and the second step of Sugiyama's framework to obtain an upward planarization algorithm [3]. Thereby, a planarization P with few crossings is computed without the need for levels.…”

Section: Real-world Graphsmentioning

confidence: 99%

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An SDP Approach to Multi-level Crossing Minimization

Chimani¹,

Hungerländer²,

Jünger³

et al. 2011

2011 Proceedings of the Thirteenth Workshop on Algorithm Engineering and Experiments (ALENEX)

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We present an approach based on semidefinite programs (SDP) to tackle the multi-level crossing minimization problem. Thereby, we are given a layered graph (i.e., the graph's vertices are assigned to multiple parallel levels) and ask for an ordering of the nodes on their levels such that, when drawing the graph with straight lines, the resulting number of crossings is minimized. Solving this step is crucial in the probably most widely used graph drawing scheme, the socalled Sugiyama framework.The problem has received a lot of attention both in the field of heuristics and exact methods. For a long time, integer linear programming (ILP) approaches were the only exact algorithms applicable at least to small graphs. Recently, SDP formulations for the special case of two levels were proposed and dominated the ILP for dense instances.In this paper, we present a new SDP formulation for the general multi-level version that, for two-levels, is even stronger than the aforementioned specialized SDP. As a sideproduct, we also obtain an SDP-based heuristic which in practice always gives (near-)optimal solutions.We conduct a large set of experiments, both on randomized and on real-world instances, and compare our approach to a state-of-the-art ILP-based branch-and-cut implementation. The SDP clearly dominates for denser graphs, while the ILP approach is usually faster for sparse instances. However, even for such sparse graphs, the SDP solves more instances to optimality than the ILP. In fact, there is no single instance the ILP solved, which the SDP did not. Overall, our experiments reveal that for sparse graphs, one should usually try to find an optimal solution with the ILP first. If this approach does not solve the instance to optimality within reasonable time, the SDP still has a good chance to do so.Being able to solve larger real-world instances than reported before, we are also able to evaluate heuristics for this problem. In this paper we do so for the traditional

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Quasi-Upward Planar Drawings of Mixed Graphs with Few Bends: Heuristics and Exact Methods

Binucci

Didimo

2014

Algorithms and Computation

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Layer-free upward crossing minimization

Cited by 23 publications

References 12 publications

Upward Planarity Testing via SAT

Upward Planarity Testing via SAT

An SDP Approach to Multi-level Crossing Minimization

Quasi-Upward Planar Drawings of Mixed Graphs with Few Bends: Heuristics and Exact Methods

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