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

Abstract

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.

Finding Optimal Sequences for Area Aggregation—A ⋆ vs. Integer Linear Programming

“…This is a typical optimization problem, where we stick to hard constraints and try to fulfill soft ones as well as possible. Optimization for map generalization is important not only because it finds optimal solutions, but also because it helps us evaluate the quality of a model [26,29,30]. When we wish to minimize the changes of types in aggregating areas one-by-one, a model could be to minimize the greatest change over all the steps.…”

Multi-page Labeling of Small-screen Maps with a Graph-coloring Approach

“…Even for the simple case that each feature has one label candidate and the labels are unit-squares, computing such an independent set is NP-hard (Fowler et al, 1981). Hence, heuristics (e.g., Christensen et al (1994)), approximation algorithms (e.g., van Kreveld et al (1999)) and exact algorithms based on integer linear programming (e.g., Haunert and Wolff (2017)) have been developed.…”

“…Finding such sets has been extensively investigated in research on label placement before; e.g., see Agarwal et al (1998). Figure 2 shows an example that we have obtained in such a way using a simple integer linear programming formulation for maximum weight independent set (Haunert and Wolff, 2017). We observe that the last three pages only contain one label each, while the first two pages are densely packed.…”

Exploring Semi-Automatic Map Labeling

“…The simplest weight function is w ≡ 1, which just counts the number of selected labels. But more advanced weight functions, defined for single labels, pairs of labels, or even larger subsets, in order to model various cartographic principles are possible [12,21,27]. We want to explore user modifications in our semi-automatic labeling process, which, for example, change the set L of label candidates to a set L or the weight function w to a function w .…”

“…[6,24,[28][29][30]; for surveys and general introductions see, e.g., [15,24,31]. More recent works introduced advanced multi-criteria optimization models [12,21,27] that can express more accurately several established cartographic principles, but still with the aim of a full automation of the map labeling process. While progress is made by incorporating more comprehensive cartographic rules for label placement, none of the above approaches includes decisions made by human experts -other than setting preferences, parameters, and priorities in the different scoring functions that control a single optimization run of the respective algorithm.…”