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An SDP approach to multi-level crossing minimization

Content Provider	ACM Digital Library
Author	Chimani, Markus Hungerlnder, Philipp Jnger, Michael Mutzel, Petra
Copyright Year	2012
Abstract	We present an approach based on semidefinite programs (SDP) to tackle the multi-level crossing minimization problem. We are given a layered graph (i.e., the graph's vertices are assigned to multiple parallel levels) and are asked for an ordering of the nodes on each level such that, when drawing the graph with straight lines, the resulting number of crossings is minimized. Solving this step is crucial in what is probably the most widely used graph drawing scheme, the Sugiyama framework. The problem has received a lot of attention in both the fields of heuristics and exact methods. For a long time, integer linear programming (ILP) approaches were the only exact algorithms applicable, at least for small graphs. Recently, SDP formulations for the special case of two levels were proposed and dominated the ILP for dense instances. In this article, 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 by-product, 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 that the ILP solved that 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 article, we do so for the traditional barycenter-heuristic (showing that it leaves a large gap to the true optimum) and the state-of-the-art upward-planarization method (showing that it is usually close to the optimum).
Starting Page	3.1
Ending Page	3.26
Page Count	26
File Format	PDF
ISSN	10846654
e-ISSN	10846654
DOI	10.1145/2133803.2330084
Journal	Journal of Experimental Algorithmics (JEA)
Volume Number	17
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2012-09-01
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Exact multi-level crossing minimization SDP approach Sugiyama Experimental evaluation
Content Type	Text
Resource Type	Article
Subject	Theoretical Computer Science

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3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
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