Ignaz Rutter

Simultaneous PQ-Ordering with Applications to Constrained Embedding Problems

In this article, we define and study the new problem of S<scp>imultaneous</scp> PQ-O<scp>rdering</scp>. Its input consists of a set of PQ-trees, which represent sets of circular orders of their leaves, together with a set of child-parent relations between these PQ-trees, such that the leaves of the child form a subset of the leaves of the parent. S<scp>imultaneous</scp> PQ-O<scp>rdering</scp> asks whether orders of the leaves of each of the trees can be chosen <i>simultaneously</i>; that is, for every child-parent relation, the order chosen for the parent is an extension of the order chosen for the child. We show that S<scp>imultaneous</scp> PQ-O<scp>rdering</scp> is <i>NP</i>-complete in general, and we identify a family of instances that can be solved efficiently, the <i>2-fixed instances</i>. We show that this result serves as a framework for several other problems that can be formulated as instances of S<scp>imultaneous</scp> PQ-O<scp>rdering</scp>. In particular, we give linear-time algorithms for recognizing simultaneous interval graphs and extending partial interval representations. Moreover, we obtain a linear-time algorithm for P<scp>artially</scp> PQ-C<scp>onstrained</scp> P<scp>lanarity</scp> for biconnected graphs, which asks for a planar embedding in the presence of PQ-trees that restrict the possible orderings of edges around vertices, and a quadratic-time algorithm for S<scp>imultaneous</scp> E<scp>mbedding with</scp> F<scp>ixed</scp> E<scp>dges</scp> for biconnected graphs with a connected intersection. Both results can be extended to the case where the input graphs are not necessarily biconnected but have the property that each cutvertex is contained in at most two nontrivial blocks. This includes, for example, the case where both graphs have a maximum degree of 5.

Manhattan-Geodesic embedding of planar graphs

In this paper, we explore a new convention for drawing graphs, the (Manhattan-) geodesic drawing convention. It requires that edges are drawn as interior-disjoint monotone chains of axis-parallel line segments, that is, as geodesics with respect to the Manhattan metric. First, we show that geodesic embeddability on the grid is equivalent to 1-bend embeddability on the grid. For the latter question an efficient algorithm has been proposed. Second, we consider geodesic point-set embeddability where the task is to decide whether a given graph can be embedded on a given point set. We show that this problem is

Computing large matchings fast

\mathcal{NP}

Search-Space size in contraction hierarchies

-hard. In contrast, we efficiently solve geodesic polygonization—the special case where the graph is a cycle. Third, we consider geodesic point-set embeddability where the vertex–point correspondence is given. We show that on the grid, this problem is

A Kuratowski-type theorem for planarity of partially embedded graphs

\mathcal{NP}

Using ILP/SAT to Determine Pathwidth, Visibility Representations, and other Grid-Based Graph Drawings

-hard even for perfect matchings, but without the grid restriction, we solve the matching problem efficiently.

Augmenting the Connectivity of Planar and Geometric Graphs

In this paper we present algorithms for computing large matchings in 3-regular graphs, graphs with maximum degree 3, and 3-connected 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 best-known algorithm for computing maximum matchings in general graphs, which runs in O(√nm) time, where n denotes the number of vertices and m the number of edges of the given graph. For the classes of 3-regular graphs and graphs with maximum degree 3 the bounds we achieve are known to be best possible. 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 3-regular graphs and 3-connected planar graphs with bounded-degree 2- and 4-block trees, respectively, we show how to compute maximum matchings in slightly superlinear time.

Simultaneous Embedding: Edge Orderings, Relative Positions, Cutvertices

Contraction hierarchies are a speed-up technique to improve the performance of shortest-path computations, which works very well in practice. Despite convincing practical results, there is still a lack of theoretical explanation for this behavior. In this paper, we develop a theoretical framework for studying search space sizes in contraction hierarchies. We prove the first bounds on the size of search spaces that depend solely on structural parameters of the input graph, that is, they are independent of the edge lengths. To achieve this, we establish a connection with the well-studied elimination game. Our bounds apply to graphs with treewidth k, and to any minor-closed class of graphs that admits small separators. For trees, we show that the maximum search space size can be minimized efficiently, and the average size can be approximated efficiently within a factor of 2. We show that, under a worst-case assumption on the edge lengths, our bounds are comparable to the recent results of Abraham et al. [1], whose analysis depends also on the edge lengths. As a side result, we link their notion of highway dimension (a parameter that is conjectured to be small, but is unknown for all practical instances) with the notion of pathwidth. This is the first relation of highway dimension with a well-known graph parameter.

Optimal Orthogonal Graph Drawing with Convex Bend Costs

A partially embedded graph (or Peg) is a triple (G,H,H), where G is a graph, H is a subgraph of G, and H is a planar embedding of H. We say that a Peg(G,H,H) is planar if the graph G has a planar embedding that extends the embedding H. We introduce a containment relation of Pegs analogous to graph minor containment, and characterize the minimal non-planar Pegs with respect to this relation. We show that all the minimal non-planar Pegs except for finitely many belong to a single easily recognizable and explicitly described infinite family. We also describe a more complicated containment relation which only has a finite number of minimal non-planar Pegs. Furthermore, by extending an existing planarity test for Pegs, we obtain a polynomial-time algorithm which, for a given Peg, either produces a planar embedding or identifies an obstruction.

Disconnectivity and relative positions in simultaneous embeddings

We present a simple and versatile formulation of grid-based graph representation problems as an integer linear program ILP and a corresponding SAT instance. In a grid-based representation vertices and edges correspond to axis-parallel boxes on an underlying integer grid; boxes can be further constrained in their shapes and interactions by additional problem-specific constraints. We describe a general d-dimensional model for grid representation problems. This model can be used to solve a variety of NP-hard graph problems, including pathwidth, bandwidth, optimum st-orientation, area-minimal bar-k visibility representation, boxicity-k graphs and others. We implemented SAT-models for all of the above problems and evaluated them on the Rome graphs collection. The experiments show that our model successfully solves NP-hard problems within few minutes on small to medium-size Rome graphs.