
Peng, D., Wolff, A., Haunert, J.H.: Using the A\($^\star$\) Algorithm to Find Optimal Sequences for Area Aggregation. In: Peterson, M.P. (ed.) Proc. 28th Int. Cartogr. Conf. (ICC'17)  Advances in Cartogr. & GIScience. p. 389404. SpringerVerlag (2017).

van Dijk, T.C., Fink, M., Fischer, N., Lipp, F., Markfelder, P., Ravsky, A., Suri, S., Wolff, A.: Block Crossings in Storyline Visualizations. Journal of Graph Algorithms & Applications. 21, 873913 (2017).
Storyline visualizations help visualize encounters of the characters in a story over time. Each character is represented by an xmonotone curve that goes from left to right visualizing progression of time. A meeting is represented by having the characters that participate in the meeting run close together for some time. In order to keep the visual complexity low, rather than just minimizing pairwise crossings of curves, we propose to count block crossings, that is, pairs of intersecting bundles of lines. In a block crossing, two blocks of parallel lines intersect each other, which is less distracting than the same number of individual crossings being spread over the drawing. In this paper, we show that minimizing the number of block crossings is NPhard, even if all meetings are of size 2. For this special case, we present a greedy heuristic, which we evaluate experimentally. We show that the general case is fixedparameter tractable. Our main results is a constantfactor approximation algorithm for meetings of bounded size. The algorithm is based on (approximately) solving a hyperedge deletion problem on hypergraphs that may be of independent interest.

Nöllenburg, M., Wolff, A.: Drawing and Labeling HighQuality Metro Maps by MixedInteger Programming. IEEE Transactions on Visualization and Computer Graphics. 17, 626641 (2011).
Metro maps are schematic diagrams of public transport networks that serve as visual aids for route planning and navigation tasks. It is a challenging problem in network visualization to automatically draw appealing metro maps. There are two aspects to this problem that depend on each other: the layout problem of finding station and link coordinates and the labeling problem of placing nonoverlapping station labels. par In this paper we present a new integral approach that solves the combined layout and labeling problem (each of which, independently, is known to be NPhard) using mixedinteger programming (MIP). We identify seven design rules used in most realworld metro maps. We split these rules into hard and soft constraints and translate them into a MIP model. Our MIP formulation finds a metro map that satisfies all hard constraints (if such a drawing exists) and minimizes a weighted sum of costs that correspond to the soft constraints. We have implemented the MIP model and present a case study and the results of an expert assessment to evaluate the performance of our approach in comparison to both manually designed official maps and results of previous layout methods.

Erlebach, T., Hagerup, T., Jansen, K., Minzlaff, M., Wolff, A.: Trimming of Graphs, with Application to Point Labeling. Theory of Computing Systems. 47, 613636 (2010).
For \($t>0$\) and \($g\ge 0$\), a vertexweighted graph of total weight \($W$\) is \emph\($(t,g)$\)trimmable if it contains a vertexinduced subgraph of total weight at least \($(11/t)W$\) and with no simple path of more than \($g$\) edges. A family of graphs is emphtrimmable if for every constant \($t>0$\), there is a constant \($g\ge 0$\) such that every vertexweighted graph in the family is \($(t,g)$\)trimmable. We show that every family of graphs of bounded domino treewidth is trimmable. This implies that every family of graphs of bounded degree is trimmable if the graphs in the family have bounded treewidth or are planar. We also show that every family of directed graphs of bounded layer bandwidth (a less restrictive condition than bounded directed bandwidth) is trimmable. As an application of these results, we derive polynomialtime approximation schemes for various forms of the problem of labeling a subset of given weighted point features with nonoverlapping sliding axesparallel rectangular labels so as to maximize the total weight of the labeled features, provided that the ratios of label heights or the ratios of label lengths are bounded by a constant. This settles one of the last major open questions in the theory of map labeling.

Rutter, I., Wolff, A.: Computing Large Matchings Fast. ACM Transactions on Algorithms. 7, article 1, 21 pages (2010).
In this paper we present algorithms for computing large matchings in 3regular graphs, graphs with maximum degree 3, and 3connected planar graphs. The algorithms give a guarantee on the size of the computed matching and take linear or slightly superlinear time. Thus they are faster than the bestknown algorithm for computing maximum matchings in general graphs, which runs in \($O(\sqrtnm)$\) time, where \($n$\) denotes the number of vertices and \($m$\) the number of edges of the given graph. For the classes of 3regular graphs and graphs with maximum degree 3, the bounds we achieve are known to be best possible. par We also investigate graphs with block trees of bounded degree, where the \($d$\)block tree is the adjacency graph of the \($d$\)connected components of the given graph. In 3regular graphs and 3connected planar graphs with boundeddegree 2 and 4block trees, respectively, we show how to compute emphmaximum matchings in slightly superlinear time.

Goaoc, X., Kratochvíl, J., Okamoto, Y., Shin, C.S., Spillner, A., Wolff, A.: Untangling a Planar Graph. Discrete & Computational Geometry. 42, 542569 (2009).
A straightline drawing \($\delta$\) of a planar graph \($G$\) need not be plane, but can be made so by emphuntangling it, that is, by moving some of the vertices of \($G$\). Let shift\($(G,\delta)$\) denote the minimum number of vertices that need to be moved to untangle \($\delta$\). We show that shift\($(G,\delta)$\) is NPhard to compute and to approximate. Our hardness results extend to a version of textsc1BendPointSetEmbeddability, a wellknown graphdrawing problem. par Further we define fix\($(G,\delta)=nshift(G,\delta)$\) to be the maximum number of vertices of a planar \($n$\)vertex graph \($G$\) that can be fixed when untangling \($\delta$\). We give an algorithm that fixes at least \($\sqrt((\log n)1)/\log \log n$\) vertices when untangling a drawing of an \($n$\)vertex graph \($G$\). If \($G$\) is outerplanar, the same algorithm fixes at least \($\sqrtn/2$\) vertices. On the other hand we construct, for arbitrarily large \($n$\), an \($n$\)vertex planar graph \($G$\) and a drawing \($\delta_G$\) of \($G$\) with fix\($(G,\delta_G) \le \sqrtn2+1$\) and an \($n$\)vertex outerplanar graph \($H$\) and a drawing \($\delta_H$\) of \($H$\) with fix\($(H,\delta_H) \le 2 \sqrtn1+1$\). Thus our algorithm is asymptotically worstcase optimal for outerplanar graphs.