Area aggregation in map generalisation by mixed-integer programming

Abstract

Topographic databases normally contain areas of different land cover classes, commonly defining a planar partition, that is, gaps and overlaps are not allowed. When reducing the scale of such a database, some areas become too small for representation and need to be aggregated. This unintentionally but unavoidably results in changes of classes. In this article we present an optimisation method for the aggregation problem. This method aims to minimise changes of classes and to create compact shapes, subject to hard constraints ensuring aggregates of sufficient size for the target scale. To quantify class changes we apply a semantic distance measure. We give a graph theoretical problem formulation and prove that the problem is NP-hard, meaning that we cannot hope to find an efficient algorithm. Instead, we present a solution by mixed-integer programming that can be used to optimally solve small instances with existing optimisation software. In order to process large datasets, we introduce specialised heuristics that allow certain variables to be eliminated in advance and a problem instance to be decomposed into independent sub-instances. We tested our method for a dataset of the official German topographic database ATKIS with input scale 1:50,000 and output scale 1:250,000. For small instances, we compare results of this approach with optimal solutions that were obtained without heuristics. We compare results for large instances with those of an existing iterative algorithm and an alternative optimisation approach by simulated annealing. These tests allow us to conclude that, with the defined heuristics, our optimisation method yields high-quality results for large datasets in modest time.

Spatial Relations Using High Level Concepts

“…Aggregation with displacement displaces the set of objects until they touch or overlap and then subsequently merges these objects [54]. If the objects in question initially touch no displacement is necessary and they can be simply merged [48,49]. Aggregation by flooding replaces the set of objects in question with a single object of greater spatial extent such as the convex hull.…”

“…Semantic constraints ensure generalization is performed in a manner which is a function of object semantics. Kieler et al [48] and Haunert and Wolff [49] propose aggregation techniques which ensure that objects merged predominantly belong to the same class. Metric constraints perform generalization in a manner which is a function of an error function.…”

Beyond Maximum Independent Set: An Extended Integer Programming Formulation for Point Labeling

“…For the aggregation of areas in map generalization, we [35] developed an ILP-based method and presented a comparison with an approach based on simulated annealing. Since the exact ILP-based method could not solve large problem instances, it was combined with a heuristic decomposition technique.…”

Partitioning Polygons via Graph Augmentation

“…Usually, every output area must have at least a certain minimal size. Subject to this requirement, Haunert and Wolff [12] suggested minimizing a cost function that combines two objectives: the overall weighted class change should be small and the resulting areas should be geometrically compact. They showed that the problem is NP-hard and developed an exact method based on integer linear programming and a heuristic method based on simulated annealing.…”