Abstract.A directed acyclic graph is upward planar if it allows a drawing without edge crossings where all edges are drawn as curves with monotonously increasing y-coordinates. The problem to decide whether a graph is upward planar or not is NP-complete in general, and while special graph classes are polynomial time solvable, there is not much known about solving the problem for general graphs in practice. The only attempt so far was a branch-and-bound algorithm over the graph's triconnectivity structure which was able to solve sparse graphs.In this paper, we propose a fundamentally different approach, based on the seemingly novel concept of ordered embeddings. We carefully model the problem as a special SAT instance, i.e., a logic formula for which we check satisfiability. Solving these SAT instances allows us to decide upward planarity for arbitrary graphs. We then show experimentally that this approach seems to dominate the known alternative approaches and is able to solve traditionally used graph drawing benchmarks effectively.

A directed acyclic graph (DAG) is upward planar if it can be drawn without any crossings while all edges-when following them in their direction-are drawn with strictly monotonously increasing ycoordinates. Testing whether a graph allows such a drawing is known to be NP-complete, but there is a substantial collection of different algorithmic approaches known in literature. In this paper, we give an overview of the known algorithms, ranging from combinatorial FPT and branch-and-bound algorithms to ILP and SAT formulations. Most approaches of the first class have only been considered from the theoretical point of view, but have never been implemented. For the first time, we give an extensive experimental comparison between virtually all known approaches to the problem. Furthermore, we present a new SAT formulation based on a recent theoretical result by Fulek et al. [8], which turns out to perform best among all known algorithms.

The upward crossing number problem asks for a drawing of the graph into the plane with the minimum number of edge crossings where the edges are drawn as monotonously increasing curves w.r.t. the y-axis. While there is a large body of work on solving this central graph drawing problem heuristically, we present the first approach to solve the problem to proven optimality. Our approach is based on a reformulation of the problem as a boolean formula that can be iteratively tightened and resolved. In our experiments, we show the practical applicability and limits of our approach. Furthermore, we can now for the first time evaluate the state-of-the-art heuristics w.r.t. true optimum solutions. This leads to the finding that these algorithms are in general surprisingly far away from the optimum. Finally, we show that we can use our approach as a strong heuristic: even after only one minute of running time, our approach typically gives better solutions than the known heuristics for medium sized instances.

A directed acyclic graph (DAG) is upward planar if it can be drawn without any crossings while all edges—when following them in their direction—are drawn with strictly monotonously increasing y -coordinates. Testing whether a graph allows such a drawing is known to be NP-complete, and while the problem is polynomial-time solvable for special graph classes, there is not much known about solving the problem for general graphs in practice . The only attempt so far has been a branch-and-bound algorithm over the graph’s triconnectivity structure, which was able to solve small graphs. Furthermore, there are some known FPT algorithms to deal with the problem. In this article, we propose two fundamentally different approaches based on the seemingly novel concept of ordered embeddings and on the concept of a Hanani-Tutte-type characterization of monotone drawings. In both approaches, we model the problem as special SAT instances, that is, logic formulae for which we check satisfiability. Solving these SAT instances allows us to decide upward planarity for arbitrary graphs. For the first time, we give an extensive experimental comparison between virtually all known approaches to the problem. To this end, we also investigate implementation issues and different variants of the known algorithms as well as of our SAT approaches and evaluate all algorithms on real-world as well as on constructed instances. We also give a detailed performance study of the novel SAT approaches. We show that the SAT formulations outperform all known approaches for graphs with up to 400 edges. For even larger graphs, a modified branch-and-bound algorithm becomes competitive.

Abstract. The equivalence problem for monotone formulae in normal form Monet is in coNP, is probably not coNP-complete [10], and is solvable in quasi-polynomial time n o(log n) [7]. We show that the straightforward reduction from Monet to UnSat yields instances, on which actual Sat-solvers (SAT4J) are slower than current implementations of Monet-algorithms [9]. We then improve these implementations of Monet-algorithms notably, and we investigate which techniques from Sat-solving are useful for Monet. Finally, we give an advanced reduction from Monet to UnSat that yields instances, on which the Sat-solvers reach running times, that seem to be magnitudes better than what is reachable with the current implementations of Monetalgorithms.