Thomas C. van Dijk

Inclusion/Exclusion Meets Measure and Conquer

Inclusion/exclusion and measure and conquer are two central techniques from the field of exact exponential-time algorithms that recently received a lot of attention. In this paper, we show that both techniques can be used in a single algorithm. This is done by looking at the principle of inclusion/exclusion as a branching rule. This inclusion/exclusion-based branching rule can be combined in a branch-and-reduce algorithm with traditional branching rules and reduction rules. The resulting algorithms can be analysed using measure and conquer allowing us to obtain good upper bounds on their running times.In this way, we obtain the currently fastest exact exponential-time algorithms for a number of domination problems in graphs. Among these are faster polynomial-space and exponential-space algorithms for #Dominating Set and Minimum Weight Dominating Set (for the case where the set of possible weight sums is polynomially bounded), and a faster polynomial-space algorithm for Domatic Number.This approach is also extended in this paper to the setting where not all requirements in a problem need to be satisfied. This results in faster polynomial-space and exponential-space algorithms for Partial Dominating Set, and faster polynomial-space and exponential-space algorithms for the well-studied parameterised problem k-Set Splitting and its generalisation k-Not-All-Equal Satisfiability.

Integer Maximum Flow in Wireless Sensor Networks with Energy Constraint

We study the integer maximum flow problem on wireless sensor networks with energy constraint. In this problem, sensor nodes gather data and then relay them to a base station, before they run out of battery power. Packets are considered as integral units and not splittable. The problem is to find the maximum data flow in the sensor network subject to the energy constraint of the sensors. We show that this integralversion of the problem is stronglyNP-complete and in fact APX-hard. It follows that the problem is unlikely to have a polynomial time approximation scheme. Even when restricted to graphs with concrete geometrically defined connectivity and transmission costs, the problem is still strongly NP-complete. We provide some interesting polynomial time algorithms that give good approximations for the general case nonetheless. For networks of bounded treewidth greater than two, we show that the problem is weaklyNP-complete and provide pseudo-polynomial time algorithms. For a special case of graphs with treewidth two, we give a polynomial time algorithm.

Simultaneous Drawing of Planar Graphs with Right-Angle Crossings and Few Bends ?

Given two planar graphs that are defined on the same set of vertices, a RAC simultaneous drawing is a drawing of the two graphs where each graph is drawn planar, no two edges overlap, and edges of one graph can cross edges of the other graph only at right angles. In the geometric version of the problem, vertices are drawn as points and edges as straight-line segments. It is known, however, that even pairs of very simple classes of planar graphs (such as wheels and matchings) do not always admit a geometric RAC simultaneous drawing.

Interactive focus maps using least-squares optimization

We present a new algorithm that enlarges a focus region in a given network map without removing non-focus (i.e., context) network parts from the map or changing the map’s size. In cartography, this problem is usually tackled with fish-eye projections, which, however, introduce severe distortion. Our new algorithm minimizes distortion and, with respect to this objective, produces results of similar quality compared to an existing algorithm. In contrast to the existing algorithm, the new algorithm achieves real-time performance that allows its application in interactive systems. We target applications where a user sets a focus by brushing parts of the network or the focus region is defined as the neighborhood of a moving user. A crucial feature of the algorithm is its capability of avoiding unwanted edge crossings. Basically, we first solve a least-squares optimization problem without constraints for avoiding edge crossings. The solution we find is then used to identify a small set of constraints needed for a crossing-free solution and, beyond this, allows us to start an animation enlarging the focus region before the final, crossing-free solution is found. Moreover, memorizing the non-crossing constraints from an initial run of the algorithm allows us to achieve a better runtime on any further run – assuming that the focus region does not move too much between two consecutive runs. As we show with experiments on real-world data, this enables response times well below 1 second.

Accentuating focus maps via partial schematization

We present an algorithm for schematized focus maps. Focus maps integrate a high detailed, enlarged focus region continuously in a given base map. Recent methods integrate both with such low distortion that the focus region becomes hard to identify. We combine focus maps with partial schematization to display distortion of the context and to emphasize the focus region. Schematization visually conveys geographical accuracy, while not increasing map complexity. We extend the focus-map algorithm to incorporate geometric proximity relationships and show how to combine focus maps with schematization in order to cater to different use cases.

How to eat a graph: computing selection sequences for the continuous generalization of road networks

In a connected weighted graph, consider deleting the edges one at a time, in some order, such that after every deletion the remaining edges are still connected. We study the problem of finding such a deletion sequence that maximizes the sum of the weights of the edges in all the distinct graphs generated: the weight of an edge is counted in every graph that it is in. This effectively asks for the high-weight edges to remain in the graph as long as possible, subject to connectivity. We apply this to road network generalization in order to generate a sequence of successively more generalized maps of a road network so that these maps go well together, instead of considering each level of generalization independently. In particular, we look at the problem of making a road segment selection that is consistent across zoom levels. We show that the problem is NP-hard and give an integer linear program (ILP) that solves it optimally. Solving this ILP is only feasible for small instances. Next we develop constant-factor approximation algorithms and heuristics. We experimentally demonstrate that these heuristics perform well on real-world instances.

Simultaneous Drawing of Planar Graphs with Right-Angle Crossings and Few Bends

Given two planar graphs that are defined on the same set of vertices, a RAC simultaneous drawing is one in which each graph is drawn planar, there are no edge overlaps and the crossings between the two graphs form right angles. The geometric version restricts the problem to straight-line drawings. It is known, however, that there exists a wheel and a matching which do not admit a geometric RAC simultaneous drawing. In order to enlarge the class of graphs that admit RAC simultaneous drawings, we allow bends in the resulting drawings. We prove that two planar graphs always admit a RAC simultaneous drawing with six bends per edge each, in quadratic area. For more restricted classes of planar graphs (i.e., matchings, paths, cycles, outerplanar graphs and subhamiltonian graphs), we manage to significantly reduce the required number of bends per edge, while keeping the area quadratic.

Active Learning for Classifying Template Matches in Historical Maps

Historical maps are important sources of information for scholars of various disciplines. Many libraries are digitising their map collections as bitmap images, but for these collections to be most useful, there is a need for searchable metadata. Due to the heterogeneity of the images, metadata are mostly extracted by hand—if at all: many collections are so large that anything more than the most rudimentary metadata would require an infeasible amount of manual effort. We propose an active-learning approach to one of the practical problems in automatic metadata extraction from historical maps: locating occurrences of image elements such as text or place markers. For that, we combine template matching (to locate possible occurrences) with active learning (to efficiently determine a classification). Using this approach, we design a human computer interaction in which large numbers of elements on a map can be located reliably using little user effort. We experimentally demonstrate the effectiveness of this approach on real-world data.

Map schematization with circular arcs

We present an algorithm to compute schematic maps with circular arcs. Our algorithm iteratively replaces two consecutive arcs with a single arc to reduce the complexity of the output map and thus to increase its level of abstraction. Our main contribution is a method for replacing arcs that meet at high-degree vertices. This allows us to greatly reduce the output complexity, even for dense networks. We experimentally evaluate the effectiveness of our algorithm in three scenarios: territorial outlines, road networks, and metro maps. For the latter, we combine our approach with an algorithm to more evenly distribute stations. Our experiments show that our algorithm produces high-quality results for territorial outlines and metro maps. However, the lack of caricature (exaggeration of typical features) makes it less useful for road networks.

Matching Labels and Markers in Historical Maps: An Algorithm with Interactive Postprocessing

In this article, we present an algorithmic system for determining the proper correspondence between place markers and their labels in historical maps. We assume that the locations of place markers (usually pictographs) and labels (pieces of text) have already been determined -- either algorithmically or by hand -- and we want to match the labels to the markers. This time-consuming step in the digitization process of historical maps is nontrivial even for humans but provides valuable metadata (e.g., when subsequently georeferencing the map). To speed up this process, we model the problem in terms of combinatorial optimization, solve that problem efficiently, and show how user interaction can be used to improve the quality of the results. We also consider a version of the model where we are given label fragments and additionally have to decide which fragments go together. We show that this problem is NP-hard. However, we give a polynomial-time algorithm for a restricted version of this fragment assignment problem. We have implemented the algorithm for the main problem and tested it on a manually extracted ground truth for eight historical maps with a combined total of more than 12,800 markers and labels. On average, the algorithm correctly matches 96% of the labels and is robust against noisy input. It furthermore performs a sensitivity analysis and in this way computes a measure of confidence for each of the matches. We use this as the basis for an interactive system where the user’s effort is directed to checking those parts of the map where the algorithm is unsure; any corrections the user makes are propagated by the algorithm. We discuss a prototype of this system and statistically confirm that it successfully locates those areas on the map where the algorithm needs help.