Beyond Maximum Independent Set: An Extended Integer Programming Formulation for Point Labeling
1
Institute of Geodesy and Geoinformation, University of Bonn, 53115 Bonn, Germany
2
Institut für Informatik, Universität Würzburg, 97074 Würzburg, Germany
*
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ISPRS Int. J. Geo-Inf. 2017, 6(11), 342; https://doi.org/10.3390/ijgi6110342
Received: 6 September 2017
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Revised: 20 October 2017
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Accepted: 25 October 2017
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Published: 6 November 2017
Map labeling is a classical problem of cartography that has frequently been approached by combinatorial optimization. Given a set of features in a map and for each feature a set of label candidates, a common problem is to select an independent set of labels (that is, a labeling without label–label intersections) that contains as many labels as possible and at most one label for each feature. To obtain solutions of high cartographic quality, the labels can be weighted and one can maximize the total weight (rather than the number) of the selected labels. We argue, however, that when maximizing the weight of the labeling, the influences of labels on other labels are insufficiently addressed. Furthermore, in a maximum-weight labeling, the labels tend to be densely packed and thus the map background can be occluded too much. We propose extensions of an existing model to overcome these limitations. Since even without our extensions the problem is NP-hard, we cannot hope for an efficient exact algorithm for the problem. Therefore, we present a formalization of our model as an integer linear program (ILP). This allows us to compute optimal solutions in reasonable time, which we demonstrate both for randomly generated point sets and an existing data set of cities. Moreover, a relaxation of our ILP allows for a simple and efficient heuristic, which yielded near-optimal solutions for our instances.
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Keywords:
map labeling; point-feature label placement; NP-hard; integer linear programming; cartographic requirements
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MDPI and ACS Style
Haunert, J.-H.; Wolff, A. Beyond Maximum Independent Set: An Extended Integer Programming Formulation for Point Labeling. ISPRS Int. J. Geo-Inf. 2017, 6, 342. https://doi.org/10.3390/ijgi6110342
AMA Style
Haunert J-H, Wolff A. Beyond Maximum Independent Set: An Extended Integer Programming Formulation for Point Labeling. ISPRS International Journal of Geo-Information. 2017; 6(11):342. https://doi.org/10.3390/ijgi6110342
Chicago/Turabian StyleHaunert, Jan-Henrik, and Alexander Wolff. 2017. "Beyond Maximum Independent Set: An Extended Integer Programming Formulation for Point Labeling" ISPRS International Journal of Geo-Information 6, no. 11: 342. https://doi.org/10.3390/ijgi6110342
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