We present algorithms for the two layer straightline crossing minimization problem that are able to compute exact optima. Our computational results lead us to the conclusion that there is no need for heuristics if one layer is fixed, even though the problem is NP-hard, and that for the general problem with two variable layers, true optima can be computed for sparse instances in which the smaller layer contains up to 15 nodes. For bigger instances, the iterated barycenter method turns out to be the method of choice among several popular heuristics whose performance we could assess by comparing their results to optimum solutions.

In a total order of the vertices of a graph, two edges with no endpoint in common can be crossing, nested, or disjoint. A k-stack (resp...

We consider graph drawings in which vertices are assigned to layers and edges are drawn as straight line-segments between vertices on adjacent layers. We prove that graphs admitting crossing-free h-layer drawings (for fixed h) have bounded pathwidth. We then use a path decomposition as the basis for a linear-time algorithm to decide if a graph has a crossing-free h-layer drawing (for fixed h). This algorithm is extended to solve a large number of related problems, including allowing at most k crossings, or removing at most r edges to leave a crossing-free drawing (for fixed k or r). If the number of crossings or deleted edges is a non-fixed parameter then these problems are NP-complete. For each setting, we can also permit downward drawings of directed graphs and drawings in which edges may span multiple layers, in which case the total span or the maximum span of edges can be minimized. In contrast to the so-called Sugiyama method for layered graph drawing, our algorithms do not assume a preassignment of the vertices to layers.

The bigraph crossing problem, embedding the two vertex sets of a bipartite graph G = (V0;V1;E) along two parallel lines so that edge crossings are minimized, has application to circuit layout and graph drawing. We consider the case where both V0 and V1 can be permuted arbitrarily -- both this and the case where the order of one vertex set is fixed are NP-hard. Two new heuristics that perform well on sparse graphs such as occur in circuit layout problems are presented. The new heuristics outperform existing heuristics on graph classes that range from application-specific to random. Our experimental design methodology ensures that differences in performance are statistically significant and not the result of minor variations in graph structure or input order.

A bipartite graph is biplanar if the vertices can be placed on two parallel lines (layers) in the plane such that there are no edge crossings when edges are drawn straight. The 2-Layer Planarization problem asks if k edges can be deleted from a given graph G so that the remaining graph is biplanar. This problem is NP-complete, as is the 1-Layer Planarization problem in which the permutation of the vertices in one layer is fixed. We give the following fixed parameter tractability results: an O(k ·6 k +|G|) algorithm for 2-Layer Planarization and an O(3 k ·|G|) algorithm for 1-Layer Planarization, thus achieving linear time for fixed k.

We give an O(ϕ k · n 2) fixed parameter tractable algorithm for the 1-SIDED CROSSING MINIMIZATION problem. The constant ϕ in the running time is the golden ratio ϕ = (1 + √ 5)/2 ≈ 1.618. The constant k is the parameter of the problem: the number of allowed edge crossings.

Abstract. Two trees with the same number of leaves have to be embedded in two layers in the plane such that the leaves are aligned in two adjacent layers. Additional matching edges between the leaves give a one-to-one correspondence between pairs of leaves of the different trees. Do there exist two planar embeddings of the two trees that minimize the crossings of the matching edges? This problem has important applications in the construction and evaluation of phylogenetic trees.

We introduce a novel decoding procedure for statistical machine translation and other ordering tasks based on a family of Very Large-Scale Neighborhoods, some of which have previously been applied to other NP-hard permutation problems. We significantly generalize these problems by simultaneously considering three distinct sets of ordering costs. We discuss how these costs might apply to MT, and some possibilities for training them. We show how to search and sample from exponentially large neighborhoods using efficient dynamic programming algorithms that resemble statistical parsing. We also incorporate techniques from statistical parsing to improve the runtime of our search. Finally, we report results of preliminary experiments indicating that the approach holds promise. 1

When dealing with a graph, any visualization strategy must rely on a layout procedure at least to initiate the process. Because the visualization process evolves within an interactive environment the choice of this layout procedure is critical and will often be based on efficiency. This paper compares two popular layout strategies, one based on the extraction of a spanning tree, the other based on edge crossing minimization of directed acyclic graphs. The comparison is made based on a large number of experimental evidence gathered through random graph generation. The main conclusion of these experiments is that, contrary to the popular belief, usage of edge crossing minimization algorithms may be extremely useful and advantageous, even under the heavy requirements of information visualization.

The barycenter heuristic is often used in practice to solve the NP-hard two-layer edge crossing minimization problem. It is well-known that the barycenter heuristic can give solutions as bad as Ω(√n) times the optimum, where n is the number of nodes in the graph. However, the example used in the proof has many isolated nodes. Makinen [8] conjectured that a better ratio bound is possible if isolated nodes are not present. We show that the ratio bound for the barycenter heuristic is still Ω(√n) even for connected bipartite graphs. We also prove a tight constant ratio bound for the barycenter heuristic on bounded-degree graphs. The bound is d - 1, where d is the maximum degree of a node in the layer that can be permuted.